Benchmark of deep-inelastic-scattering structure functions at $\mathcal {O}\left( \alpha _s ^3\right)$

Abstract

We present a benchmark comparison of the massless inclusive deep-inelastic-scattering (DIS) structure functions up to $\mathcal {O}(\alpha _s ^3)$ in perturbative QCD. The comparison is performed using the codes apfel++ and hoppet within the framework of the variable-flavour-number scheme and over a broad kinematic range relevant to the extraction of parton distribution functions. We provide results for both the single structure functions and the reduced cross sections in both neutral- and charged-current DIS. Look-up tables for future reference are included, and we also release the code used for the benchmark.

1 Introduction

Deep inelastic scattering (DIS) is one of the best theoretically understood processes in perturbative QCD, cf. Refs. [1, 2] for a review. Most significantly, it offers unique insight into the proton structure, and to this day legacy DIS data still has a major impact on fits of parton distribution functions (PDFs) [3,4,5,6,7,8,9].

The inclusive DIS cross sections can be parametrised in terms of structure functions. These structure functions are inherently non-perturbative quantities, but they can be expressed as convolutions between hard perturbative coefficient functions and PDFs, where the latter encompass the non-perturbative contribution. As of today, the massless DIS coefficient functions are fully known up to $\mathcal {O}(\alpha _s ^3)$, allowing us to compute structure functions up to next-to-next-to-next-to-leading order (N$^3$LO) accuracy [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26].¹ Very recently the code yadism [27], which computes DIS structure functions at this order, became available. However, their implementation was directly compared to apfel++ and therefore relies implicitly on the benchmark presented here.

In this paper, we present a comparison of the structure functions obtained using the two publicly available codes apfel++ [28, 29] and hoppet [30].² The benchmark that we present here serves two purposes. First, it validates the correctness of the structure functions as implemented in the two programs. This is a highly non-trivial check in that, although the coefficient functions are identical in the two programs, the underlying technologies adopted by the two codes are not. Secondly, the results presented here provide a reference for any future numerical implementation of the DIS structure functions. To make the benchmark as resilient towards the future as possible, we carry out the benchmark using the same PDF initial conditions as those used in Ref. [36]. This avoids complications caused by numerical artefacts related to pre-computed interpolation grids such as those released through the LHAPDF interface [37]. Moreover, this guarantees the independence of the benchmark from the availability of a specific PDF set. The benchmark is carried out both on the single structure functions and on the reduced cross sections, which is what is often measured in experiments.

The paper is structured as follows. In Sect. 2, we review the DIS process and define the structure functions. The numerical setup of the benchmark is presented in Sect. 3. We finally present the benchmark in Sect. 4 before concluding in Sect. 5. Appendix A contains some comments on the large-y behaviour of the non-singlet N$^3$LO coefficient functions, while Appendix B presents look-up benchmark tables for all of the structure functions at NLO, NNLO, and N$^3$LO accuracy for different values of Q and for Bjorken $x_\textrm{B}\in [10^{-5}:0.9]$.

2 The DIS structure functions

Let us begin by recalling the kinematics of the DIS process. This process is the inclusive scattering of a lepton $\ell$ with momentum $k_i$ off a proton p with momentum P via the exchange of a virtual electroweak gauge boson V with momentum q and large (negative) virtuality $Q^2=-q^2\gg \Lambda _\textrm{QCD}^2$, where $\Lambda _\textrm{QCD}$ is the typical hadronic scale. Due to the large virtuality of the vector boson, the proton breaks up leaving in the final state the scattered lepton $\ell '$ with momentum $k_f=k_i-q$ and a remnant X, with respect to which we are fully inclusive:

$$\begin{aligned} \ell (k_i)+p(P)\rightarrow V(q) \rightarrow \ell '(k_f)+X. \end{aligned}$$

(1)

It is useful to introduce the customary DIS variables $x_\textrm{B}$ (Bjorken’s variable) and $y$ (inelasticity) defined as:

$$\begin{aligned} x_\textrm{B}= \frac{Q^2}{2 P \cdot q}, \quad y= \frac{P \cdot q}{P \cdot k_i}=\frac{Q^2}{x_\textrm{B}s}, \end{aligned}$$

(2)

where $s=(k_i+P)^2$ is the collision center-of-mass energy squared.³ In order to describe the interaction between the proton and the appropriate electroweak current, i.e. the neutral current (NC) mediated by a $V=\gamma /Z$ boson or the charged current (CC) mediated by a $V=W^\pm$ boson, it is useful to consider the hadronic tensor $W_{\mu \nu }^V$. The spin-averaged hadronic tensor defines the structure functions $F_1$, $F_2$, and $F_3$ through [38]:

$$\begin{aligned} W^V_{\mu \nu }&= \left( -g_{\mu \nu } + \frac{q_\mu q_\nu }{q^2}\right) F^V_1(x_\textrm{B},Q^2)\nonumber \\&\quad + \frac{\hat{P}_\mu \hat{P}_\nu }{P\cdot q}F^V_2(x_\textrm{B},Q^2) - i\epsilon _{\mu \nu \alpha \beta }\frac{q^\alpha P^\beta }{2P\cdot q}F^V_3(x_\textrm{B},Q^2), \end{aligned}$$

(3)

where:

$$\begin{aligned} \hat{P}_\mu = P_\mu - \frac{P\cdot q}{q^2}q_\mu \,. \end{aligned}$$

(4)

It is also customary to define the longitudinal structure function as $F_L^V = F_2^V - 2x_\textrm{B}F_1^V$. With the structure functions at hand, we can write the NC cross section ($\ell ^\pm p \rightarrow \ell ^\pm + X$) as:

$$\begin{aligned} \frac{\textrm{d}\sigma _{\textrm{NC}}^\pm }{\textrm{d}x_\textrm{B}\textrm{d}Q^2} = \frac{2\pi \alpha ^2}{x_\textrm{B}Q^4}y_+ \left[ F_2^{\gamma /Z} \mp \frac{y_-}{y_+} x_\textrm{B}F_3^{\gamma /Z} - \frac{y^2}{y_+} F_L^{\gamma /Z} \right] , \nonumber \\ \end{aligned}$$

(5)

where $y_\pm =1\pm (1-y)^2$ and $\alpha$ is the fine structure constant. Similarly, the CC cross section for $\ell ^\pm p \rightarrow \nu _\ell (\overline{\nu }_\ell ) + X$ reads:

$$\begin{aligned} \frac{\textrm{d}\sigma _{\textrm{CC}}^\pm }{\textrm{d}x_\textrm{B}\textrm{d}Q^2}= & {} \frac{\pi \alpha ^2}{8 x_\textrm{B}\sin ^4\theta _W }\left( \frac{1}{M_W^2 + Q^2}\right) ^2 \nonumber \\{} & {} \times \left[ y_+ F_2^{W^{\pm }} \mp y_- x_\textrm{B}F_3^{W^{\pm }} - y^2 F_L^{W^{\pm }}\right] ,\, \end{aligned}$$

(6)

where $M_W$ is the mass of the W boson and $\theta _W$ is the electroweak mixing angle. Some experiments, such as those at the HERA collider [7], release the so-called “reduced” cross sections that are related to the standard cross sections as follows:

$$\begin{aligned} \sigma _\textrm{NC, red}^{\pm }= & {} \left[ \frac{2\pi \alpha ^2}{x_\textrm{B}Q^4}y_+\right] ^{-1}\frac{\textrm{d}\sigma _{\textrm{NC}}^\pm }{\textrm{d}x_\textrm{B}\textrm{d}Q^2}\nonumber \\= & {} F_2^{\gamma /Z} \mp \frac{y_-}{y_+} x_\textrm{B}F_3^{\gamma /Z} - \frac{y^2}{y_+} F_L^{\gamma /Z},\nonumber \\ \sigma _\textrm{CC, red}^{\pm }= & {} \left[ \frac{\pi \alpha ^2}{4\sin ^4\theta _W x_\textrm{B}}\left( \frac{1}{M_W^2 + Q^2}\right) ^2\right] ^{-1}\nonumber \\ \frac{\textrm{d}\sigma _{\textrm{CC}}^\pm }{\textrm{d}x_\textrm{B}\textrm{d}Q^2}= & {} \frac{y_+}{2} F_2^{W^{\pm }} \mp \frac{y_-}{2} x_\textrm{B}F_3^{W^{\pm }} - \frac{y^2}{2} F_L^{W^{\pm }}. \end{aligned}$$

(7)

The QCD collinear factorisation theorem allows us to express the structure functions as convolutions of the PDFs, $f_a$, with the short-distance coefficient functions, $C_i$⁴:

$$\begin{aligned} F_i^V\equiv & {} F_i^V(x_\textrm{B}, Q^2) = (x_\textrm{B})\sum _{a} \left[ C_{i,a}^{V} \otimes f_a \right] \nonumber \\{} & {} \times \left( x_\textrm{B},Q^2,\mu _\textrm{R}^2,\mu _\textrm{F}^2\right) ,\quad i=(2),(L),3, \end{aligned}$$

(8)

where the index a runs over the gluon ($a=g$) and all active quark flavours and anti-flavours ($a=q,\overline{q}$) at the scale $Q^2$.⁵ The Mellin-convolution symbol $\otimes$ implies one of the following equivalent integrals:

$$\begin{aligned} \begin{array}{rcl} &{}&{} \left[ C_{i,a}^{V} \otimes f_a\right] \left( x_\textrm{B},Q^2,\mu _\textrm{R}^2,\mu _\textrm{F}^2\right) \\ &{}&{}\quad = \int _0^1dy\int _0^1dz\, \delta (x_\textrm{B}-yz)C_{i,a}^V \nonumber \\ &{}&{}\qquad \times \left( y,\alpha _s(\mu _\textrm{R}),\frac{\mu _\textrm{R}^2}{Q^2},\frac{\mu _\textrm{F}^2}{Q^2}\right) f_a(z,\mu _\textrm{F})\\ &{}&{}\quad = \int _{x_\textrm{B}}^1 \frac{dy}{y}C_{i,a}^V\left( y,\alpha _s(\mu _\textrm{R}),\frac{\mu _\textrm{R}^2}{Q^2},\frac{\mu _\textrm{F}^2}{Q^2}\right) f_a\left( \frac{x_\textrm{B}}{y},\mu _\textrm{F}\right) \\ &{}&{}\quad = \int _{x_\textrm{B}}^1 \frac{dz}{z}C_{i,a}^V\left( \frac{x_\textrm{B}}{z},\alpha _s(\mu _\textrm{R}),\frac{\mu _\textrm{R}^2}{Q^2},\frac{\mu _\textrm{F}^2}{Q^2}\right) f_a(z,\mu _\textrm{F}). \end{array}\nonumber \\ \end{aligned}$$

(9)

2.1 Renormalisation and factorisation scale dependence

In this section, we derive the explicit dependence of the coefficient functions on the arbitrary renormalisation and factorisation scales $\mu _\textrm{R}$ and $\mu _\textrm{F}$, respectively. The coefficient function $C_{i,a}^{V}$ is an explicit function of the strong coupling $\alpha _s(\mu _\textrm{R})$ that admits the perturbative expansion:

$$\begin{aligned}{} & {} C_{i,a}^V\left( y,\alpha _s(\mu _\textrm{R}),\frac{\mu _\textrm{R}^2}{Q^2},\frac{\mu _\textrm{F}^2}{Q^2}\right) \nonumber \\{} & {} \quad =\sum _{n=0}^{\infty }\left( \frac{\alpha _s(\mu _\textrm{R})}{4\pi }\right) ^n C_{i,a}^{V,[n]}\left( y,\frac{\mu _\textrm{R}^2}{Q^2},\frac{\mu _\textrm{F}^2}{Q^2}\right) . \end{aligned}$$

(10)

Under the assumption $\alpha _s(\mu _\textrm{R})\ll 1$, the sum on the r.h.s. can be truncated to order k to obtain a N$^k$LO approximation of the structure functions.⁶$C_{i,a}^{V}$ also depends on the ratios $\mu _\textrm{R}^2/Q^2$ and $\mu _\textrm{F}^2/Q^2$. Despite that the scales $\mu _\textrm{R}$ and $\mu _\textrm{F}$ are in principle arbitrary, in a fixed-order calculation where the series in Eq. (10) includes a finite number of terms, the presence of logarithms of these ratios requires these scales to be of order $Q=\sqrt{Q^2}$. In this way, the ratios $\mu _\textrm{R}^2/Q^2$ and $\mu _\textrm{F}^2/Q^2$ are both of order one and do not compromise the convergence of the perturbative series. Variations of $\mu _\textrm{R}$ and $\mu _\textrm{F}$ around Q by a modest factor, typically of two, are often used as a proxy to estimate the possible impact of unknown higher-order corrections. This is due to the fact that any variation of $\mu _\textrm{R}$ and $\mu _\textrm{F}$ is compensated order by order in $\alpha _s$ by the evolution of strong coupling and PDFs, that in turn obey their own renormalisation group equations (RGEs):

$$\begin{aligned} \begin{array}{lll} \frac{d\ln \alpha _s(\mu _\textrm{R})}{d\ln \mu _\textrm{R}^2}&{}=&{} \beta (\alpha _s(\mu _\textrm{R}^2))=-\sum _{n=0}^{\infty }\left( \frac{\alpha _s(\mu _\textrm{R})}{4\pi }\right) ^{n+1}\beta _n,\\ \\ \frac{df_a(x,\mu _\textrm{F})}{d\ln \mu _\textrm{F}^2}&{}=&{} \sum _{b}[P_{ab}\otimes f_b](x,\mu _\textrm{F})\\ &{}=&{}\sum _{b}\sum _{n=0}^{\infty }\left( \frac{\alpha _s(\mu _\textrm{F})}{4\pi }\right) ^{n+1}[P_{ab}^{[n]}\otimes f_b](x,\mu _\textrm{F}).\\ \end{array} \end{aligned}$$

(11)

The RGEs for PDFs are usually referred to as DGLAP equations [42,43,44,45] and the kernels $P_{ab}$ are called splitting functions. Given the appropriate set of boundary conditions, the solutions of these RGEs determine the evolution (or running) of both $\alpha _s$ and $f_a$ to any scale. Similarly to the DIS coefficient functions, the $\beta$-function and the splitting functions $P_{ab}$ are expandable in powers of $\alpha _s$.

When computing a DIS structure function at N$^k$LO accuracy, it is customary to truncate also the perturbative expansions in Eq. (11) at the same relative order k. However, this is mostly a conventional procedure that is not strictly mandatory for a correct counting of the perturbative accuracy. Strictly speaking, the truncation of the expansion in Eq. (10) is responsible for the fixed-order accuracy, while the truncation of the expansions Eq. (11) defines the logarithmic accuracy. The fixed-order accuracy counts how many corrections proportional to an integer non-negative power of $\alpha _s$ are included exactly in the coefficient functions. In the DIS case, the $\mathcal {O}(\alpha _s^0)$ contribution gives leading-order (LO) accuracy, the inclusion of the $\mathcal {O}(\alpha _s)$ corrections gives next-to-leading-order (NLO) accuracy, and so on. The logarithmic accuracy instead counts the number of all-order towers of logarithms of the kind $\alpha _s^mL^n$, with $L=\ln \left( \mu _\textrm{R,F}/Q_0\right)$ and $Q_0$ the boundary-condition scale, that are being resummed by means of the evolution of $\alpha _s$ and PDFs.⁷ Leading-logarithmic (LL) accuracy is achieved summing all $\alpha _s^nL^n$ terms, next-to-leading-logarithmic (NLL) accuracy requires summing all $\alpha _s^{n+1}L^n$ terms, and so on. It is worth noting that the distinction between fixed-order and logarithmic counting is effective only when $\alpha _sL \sim 1$, i.e. when L is large enough to compensate for the assumed smallness of $\alpha _s$. This holds when $\mu _\textrm{R,F}\gg Q_0$ (or $\mu _\textrm{R,F}\ll Q_0$), which is often the case in phenomenological applications.

Although fixed-order and logarithmic accuracies have two different origins, they are not entirely unrelated. Indeed, loosely speaking, the summation of logarithms also contributes to the fixed-order counting; for instance the term $\alpha _sL$, which belongs to the LL tower, can also be regarded as a NLO contribution to the DIS structure functions. Therefore, NLO accuracy must come with at least LL resummation. In general, N$^k$LO accuracy for the DIS coefficient functions requires a minimal resummation accuracy of N$^{k-1}$LL. However, it is not incorrect to match N$^k$LO coefficient functions to N$^{k}$LL $\alpha _s$ and PDF resummation: this is what conventional N$^k$LO accuracy for structure functions prescribes.

In the following, we will adopt the “conventional” counting for the computation of the DIS structure functions up to NNLO, i.e. we will match N$^k$LO coefficient functions to N$^{k}$LL resummation, with $k=0,1,2$. At N$^3$LO, we will instead use the “minimal” prescription and match N$^3$LO coefficient functions to NNLL resummation. The reason for this choice is that, as of today, splitting functions are fully known only up to $\mathcal {O}(\alpha _s ^3)$ [17, 18, 25, 46,47,48,49,50,51,52,53,54,55]. Accompanied by the $\mathcal {O}(\alpha _s ^3)$ corrections to the $\beta$-function in Eq. (11) [56,57,58,59,60] and the mass threshold corrections to both the running coupling and the parton distributions [61,62,63], this allows us to achieve plain NNLL resummation. While the $\mathcal {O}(\alpha _s ^4)$ corrections to the $\beta$-function are known [64, 65], this is not the case for the splitting functions, which prevents attaining exact N$^3$LL resummation, in spite of the recent significant progress made in determining them at this order [66,67,68,69,70,71,72,73,74,75]. In contrast, all mass threshold corrections to the running coupling [61] and the parton distributions are known at this order [76,77,78,79,80,81,82,83,84,85,86].

It is also worth mentioning that, in order to compute the single ingredients of the factorisation formula for the structure functions on the r.h.s. of Eq. (8), it is necessary to specify a renormalisation/factorisation scheme. While the dependence on the scheme cancels out order by order in $\alpha _s$, it determines the specific form of the coefficient functions and of the anomalous dimensions ($\beta$-function and splitting functions). Throughout this work, we will use the modified minimal-subtraction ($\overline{\text{ MS }}$) scheme.

The evolution equations in Eq. (11) allow us to determine the dependence on $\mu _\textrm{R}$ and $\mu _\textrm{F}$ of the perturbative coefficients $C_{i,a}^{V,[n]}$ in Eq. (10). Indeed, provided that $\mu _\textrm{R,F}\simeq Q$, one can perturbatively solve the RGEs in Eq. (11) to evolve $\alpha _s$ and PDFs from the scales $\mu _\textrm{R}$ and $\mu _\textrm{F}$, respectively, to the scale Q [15, 87, 88], obtaining:

$$\begin{aligned} \frac{\alpha _s (Q)}{4\pi }= & {} \frac{\alpha _s ({\mu _{\textrm{R}}})}{4\pi } + \left( \frac{\alpha _s ({\mu _{\textrm{R}}})}{4\pi }\right) ^2 L_{R}\beta _0\nonumber \\{} & {} + \left( \frac{\alpha _s ({\mu _{\textrm{R}}})}{4\pi }\right) ^3 (L_{R}^2\beta _0^2 + L_{R}\beta _1) + \mathcal {O}(\alpha _s ^4), \end{aligned}$$

(12)

and:

$$\begin{aligned} \begin{array}{lll} f_a(x,Q) &{} =&{} f_a(x, {\mu _{\textrm{F}}}) - L_{F} \Bigg \{\left( \frac{\alpha _s ({\mu _{\textrm{R}}})}{4\pi }\right) P_{ab}^{[0]} \\ \\ &{} &{}+ \left( \frac{\alpha _s ({\mu _{\textrm{R}}})}{4\pi }\right) ^2 \Bigg [ P_{ab}^{[1]} - \frac{1}{2} L_{F} P_{ac}^{[0]}\otimes P_{cb}^{[0]}\\ &{}&{}- \left( \frac{L_{F}}{2} - L_{R}\right) \beta _0 P_{ab}^{[0]} \Bigg ] \\ \\ &{}&{} + \left( \frac{\alpha _s ({\mu _{\textrm{R}}})}{4\pi }\right) ^3 \Bigg [ P_{ab}^{[2]} - \frac{1}{2} L_{F} \left( P_{ac}^{[0]} \otimes P_{cb}^{[1]}\right. \\ &{}&{}\left. + P_{ac}^{[1]}\otimes P_{cb}^{[0]}\right) + \frac{1}{6} L_{F}^2 P_{ac}^{[0]}\otimes P_{cd}^{[0]}\otimes P_{db}^{[0]} \\ \\ &{} &{}+ \left( \frac{L_{F}}{2} - L_{R}\right) \beta _0 \left( L_{F} P_{ac}^{[0]}\otimes P_{cb}^{[0]}- 2 P_{ab}^{[1]}\right) \\ &{}&{}+ \left( L_{R}^2 - L_{F} L_{R} + \frac{1}{3} L_{F}^2\right) \beta _0^2 P_{ab}^{[0]} \\ \\ &{}&{}- \left( \frac{L_{F}}{2} - L_{R}\right) \beta _1 P_{ab}^{[0]} \Bigg ] \Bigg \}\otimes f_b(x, {\mu _{\textrm{F}}}) +\mathcal {O}(\alpha _s^4), \end{array}\nonumber \\ \end{aligned}$$

(13)

where a summation over repeated indices is understood and we introduced the shorthand notation:

$$\begin{aligned} L_{R} = \ln \left( \frac{\mu _\textrm{R}^2}{Q^2}\right) ,\quad L_{F} = \ln \left( \frac{\mu _\textrm{F}^2}{Q^2}\right) . \end{aligned}$$

(14)

With these equalities at hand, one can solve iteratively order by order in $\alpha _s$ the following equality:

$$\begin{aligned}{} & {} \sum _{a} \left[ C_{i,a}^{V} \otimes f_a \right] \left( x_\textrm{B},Q^2,\mu _\textrm{R}^2,\mu _\textrm{F}^2\right) \nonumber \\{} & {} \quad =\sum _{a} \left[ C_{i,a}^{V} \otimes f_a \right] \left( x_\textrm{B},Q^2,Q^2,Q^2\right) , \end{aligned}$$

(15)

which immediately implies:

(16)

where we have defined:

$$\begin{aligned} c_{i,a}^{V,[n]}(y) = C_{i,a}^{V,[n]}\left( y,1,1\right) . \end{aligned}$$

(17)

Equation (16) allows us to compute the structure functions up to N$^3$LO accuracy for any choice of renormalisation and factorisation scales in the vicinity of Q.

2.2 Characteristics of the structure functions

We now move to characterising the structure functions. In the NC case, structure functions have the following structure:

$$\begin{aligned} \begin{array}{l} F_i^{\gamma /Z} = x_\textrm{B}\sum _{a}B_a\Bigg [C_{i,\mathrm NS}^+\otimes f_{a}^{+}+C_{i,\mathrm PS}\otimes f_\textrm{PS}+C_{i,g}\otimes f_{g}\Bigg ],\\ i=2,L,\\ F_3^{\gamma /Z}= \sum _{a}D_a\left[ C_{3,\mathrm NS}^-\otimes f_{a}^{-}+C_{3,\mathrm PV}\otimes f_\textrm{PV}\right] , \end{array} \end{aligned}$$

(18)

where we have defined the following combinations of quark PDFs:

$$\begin{aligned} f_{a}^{\pm } = f_a \pm f_{\bar{a}},\quad f_{\text {PS}} = \sum _{a}f_{a}^{+}, \quad f_{\text {PV}} = \sum _{a}f_{a}^{-}. \end{aligned}$$

(19)

Importantly, in the decomposition in Eq. (18) the coefficient functions are independent of the flavour index a. Conversely, the electroweak charges $B_a$ and $D_a$ do depend on the flavour index as follows:

$$\begin{aligned} \begin{array}{l} B_a=e_a^2-2e_aV_\ell V_aP_Z+(V_\ell ^2+A_\ell ^2) (V_a^2+A_a^2)P_Z^2,\\ \\ D_a=-2e_aA_\ell A_aP_Z+4V_\ell A_\ell V_aA_aP_Z^2, \end{array} \end{aligned}$$

(20)

where:

$$\begin{aligned} P_Z=\frac{1}{4\sin ^2\theta _{W}\cos ^2\theta _{W}}\left( \frac{Q^2}{Q^2+M_Z^2}\right) , \end{aligned}$$

(21)

and $\ell$ is the lepton off which the proton scatters. The electric, vector, and axial charges for quarks and leptons are given in the Table 1.

Table 1 Electric, vector, and axial couplings for up-type, down-type, charged leptons, and neutrinos

We now move to the CC structure functions whose factorised expression reads:

$$\begin{aligned} \begin{array}{l} F_i^{W^\pm } = \frac{1}{2}x_\textrm{B}\left[ \mp C_{i,\mathrm NS}^-\otimes \delta f_\textrm{PV}+(C_{i,\mathrm NS} + C_{i,\mathrm PS})\otimes f_\textrm{PS}\right. \\ \left. +C_{i,g}\otimes f_{g} \right] ,\quad i=2,L,\\ F_3^{W^\pm }= \frac{1}{2}\left[ \pm C_{3,\mathrm NS}^+\otimes \delta f_\textrm{PS}+C_{3,\mathrm PV}\otimes f_\textrm{PV}\right] , \end{array} \end{aligned}$$

(22)

where we have defined the additional quark-PDF combinations:

$$\begin{aligned} \delta f_\textrm{PS}= & {} \sum _{a\in u\text {-type}} f_a^+ -\sum _{a\in d-\text {type}} f_a^+,\nonumber \\ \delta f_\textrm{PV}= & {} \sum _{a\in u\text {-type}} f_a^- -\sum _{a\in d-\text {type}} f_a^-. \end{aligned}$$

(23)

We notice that the expressions in Eq. (22) have been obtained under the assumption of a CKM matrix equal to the $3\times 3$ unity,⁸ which we will also use in the numerical results presented below. The corresponding expressions for a generic CKM matrix are considerably more complicated and we do not present them here.

Fig. 1

The structure functions $F_1$ (top row), $F_2$ (middle row), and $F_3$ (bottom row) for NC (left column), positive CC (middle column), and negative CC (right column) at N$^3$LO as functions of $x_\textrm{B}$ spanning between $10^{-5}$ and 0.9 and for various values of the energy Q. The lower panels show the ratio between apfel++ and hoppet

Fig. 2

The reduced cross sections $\sigma _\textrm{NC, red}^{+}$ (top left), $\sigma _\textrm{NC, red}^{-}$ (top right), $\sigma _\textrm{CC, red}^{+}$ (bottom left) and $\sigma _\textrm{NC, red}^{-}$ (bottom right) at N$^3$LO as functions of $x_\textrm{B}$ spanning between $10^{-5}$ and 0.9 and for various values of the energy Q. The centre-of-mass energy is set to $\sqrt{s}=320$ GeV. The lower panels show the ratio between apfel++ and hoppet

Fig. 3

The structure function $F_2^\textrm{NC}$ plotted as a function of $x_\textrm{B}$ in the range $[10^{-5}:0.9]$ at $Q=2$ GeV (left) and $Q=10$ GeV (right). Each plot displays the curves at LO, NLO, NNLO, and N$^3$LO with the lower panel showing the ratio to N$^3$LO

3 Numerical setup

For our benchmark, rather than relying on a set of tabulated PDFs as for example delivered by LHAPDF [37], we decided to use a set of realistic initial-scale conditions having a simple analytic form and to carry out the evolution ourselves. Besides the obvious advantage of having full numerical control on our results, we believe that this choice will allow for easier comparison to our benchmark results. To this purpose, we selected as initial conditions for the evolution the parameterisation of Sect. 1.3 of Ref. [36]. Specifically, we chose $Q_0 = \sqrt{2} \,\textrm{GeV}$ as an initial scale with $\alpha _s (Q_0) = 0.35$. At the initial scale $Q_0$, only gluon and up, down, and strange quark PDFs are present while charm, bottom, and top quark PDFs are assumed to be identically zero and have their production thresholds at $m_c=(\sqrt{2}+\epsilon )$ GeV,⁹$m_b= 4.5$ GeV, and $m_t= 175$ GeV, respectively. At the initial scale $Q_0$, the PDFs are given by:

$$\begin{aligned} x u_v(x, Q_0)&= 5.107200\,x^{0.8} (1-x)^3\,, \end{aligned}$$

(24a)

$$\begin{aligned} x d_v(x, Q_0)&= 3.064320\,x^{0.8} (1-x)^4\,,\end{aligned}$$

(24b)

$$\begin{aligned} x{\bar{d}}(x, Q_0)&= 0.1939875\,x^{-0.1} (1-x)^6\,,\end{aligned}$$

(24c)

$$\begin{aligned} x{\bar{u}}(x, Q_0)&= x{\bar{d}}(x,Q_0) (1-x)\,,\end{aligned}$$

(24d)

$$\begin{aligned} x s(x, Q_0)&= x{\bar{s}}(x, Q_0) = 0.2\,(x{\bar{d}}(x, Q_0) + x{\bar{u}}(x, Q_0))\,,\end{aligned}$$

(24e)

$$\begin{aligned} x g(x, Q_0)&= 1.7\,x^{-0.1} (1-x)^5\,, \end{aligned}$$

(24f)

where the valence distributions are defined as $u_v \equiv u - \bar{u}$ and $d_v \equiv d - {\bar{d}}$. We carry out the evolution in the variable-flavour-number scheme, that is by including quark-mass thresholds in both coupling and the PDF evolutions. The resulting set of evolved PDFs both from apfel++ and hoppet are in perfect agreement with the tables in Ref. [36] at all perturbative orders.

Fig. 4

Relative scale variations on $F_2^\textrm{NC}$ as functions of $x_\textrm{B}$ for $Q=2$ GeV (left) and $Q=10$ GeV (right). The red (blue) bands correspond to variations of $\mu _\textrm{R}$ ($\mu _\textrm{F}$) by a factor of 2 up and down around Q. Variations are shown for NLO (upper panels), NNLO (central panels), and N$^3$LO (lower panels)

Starting from NNLO accuracy, both splitting functions and coefficient functions become analytically very convoluted and it is customary to resort to the parameterisations provided by Moch, Vermaseren and Vogt [14,15,16,17,18,19,20,21,22,23,24]. These parameterisations are expected to agree with their exact counterparts at the level of $10^{-4}$ relative accuracy. In this benchmark, we thus employ exact splitting and coefficient functions up to NLO, while we use the parameterisations beyond,¹⁰ For the PDF mass threshold corrections we use the exact expressions at all orders. In Appendix A, we comment on the differences between exact and parametrised coefficient functions.

Finally, we point out that the NC structure functions also depend on the weak mixing angle (see Sect. 2.2). In this benchmark, we employ the leading-order relation $\sin ^2\theta _W = 1 - \frac{M_W^2}{M_Z^2}$ to compute it, using $M_W=80.377$ GeV and $M_Z=91.1876$ GeV for the mass values of W and Z bosons, respectively.

4 Benchmark results

In this section, we present the results of the benchmark between apfel++ and hoppet. Here, we will assess the level of agreement between the two codes at N$^3$LO accuracy by means of a set of plots. The excellent agreement found at this order (see below) immediately implies that the agreement at lower orders is at least as good. In this section, we also take the chance to discuss the impact of scale variations on DIS structure functions at the available perturbative orders, as well as the degree of perturbative convergence moving from LO to N$^3$LO. In Appendix B, instead, we provide look-up tables with predictions at all available perturbative orders over a broad kinematic range and for all of the DIS structure functions. We point out that apfel++ and hoppet are in exact agreement within the digits shown in those tables. Therefore, they can be used as a reference for future numerical implementations of the DIS structure functions. We also release the code used to produce them (see Appendix B for details).

In Fig. 1, we show all structure functions both in the NC and in the CC channels at N$^3$LO over a wide kinematic range in $x_\textrm{B}$ and Q. Here, we set $\mu _\textrm{R}=\mu _\textrm{F}=Q$. While the upper panel of each plot displays the absolute values of the structure functions, the lower panel shows the ratio between apfel++ and hoppet. It is evident that the agreement between the two codes is excellent all across the board. Specifically, we observe that the relative accuracy is well below $10^{-5}$ everywhere, except for $F_3^{W^-}$ at around $x_\textrm{B}\sim 3\cdot 10^{-3}-10^{-2}$. However, this slight degradation in relative accuracy is due to the fact that $F_3^{W^-}$ changes sign in that region.

As discussed in Sect. 2, it is also relevant to consider the DIS reduced cross sections defined in Eq. (7). As a matter of fact, the HERA collider has delivered measurements for these observables [7] that are currently being employed in most of the modern PDF determinations [4,5,6]. In Fig. 2, we show N$^3$LO predictions for NC (top row) and CC (bottom row) reduced cross sections relevant to $e^+p$ (left column) and $e^-p$ (right column) collisions. The centre-of-mass energy is set to $\sqrt{s}=320$ GeV, close to that of the latest runs of HERA. A broad kinematic range in $x_\textrm{B}$ and Q is covered and again we set $\mu _\textrm{R}=\mu _\textrm{F}=Q$. We notice that the curves, presented as functions of $x_\textrm{B}$ for different values of Q, are limited in $x_\textrm{B}$ by the physical requirement on the inelasticity $y\le 1$ (see Eq. (2)). As above, the lower panel of each plot shows the ratio between predictions obtained with apfel++ and hoppet. As expected from the results presented in Fig. 1, the two codes agree well within $10^{-5}$ relative accuracy over the full kinematic range also for the reduced cross sections.

Having at our disposal four consecutive perturbative orders, it is interesting to study how well the QCD perturbative series converges in the case of inclusive DIS structure functions. In Fig. 3, we display $F_2^\textrm{NC}$ as a function of $x_\textrm{B}$ at $Q=2$ GeV (left) and $Q=10$ GeV (right) computed with $\mu _\textrm{R}=\mu _\textrm{F}=Q$. Each plot shows this structure function for all perturbative orders between LO and N$^3$LO, with the lower panel giving the ratio to N$^3$LO. The pattern is somewhat the expected one. At $Q=2$ GeV, due to the relatively large value of $\alpha _s$, the convergence is slower with differences between NNLO and N$^3$LO that can exceed 10%, particularly at small and large values of $x_\textrm{B}$. At $Q=10$ GeV, instead, the convergence is much faster with NNLO and N$^3$LO very close to each other everywhere, except for very large values of $x_\textrm{B}$.

The perturbative convergence can also be studied by looking at how renormalisation and factorisation scale variations behave. In Sect. 2, we provided all relevant expressions to perform these variations up to N$^3$LO accuracy. We point out that we have checked that apfel++ and hoppet agree within the same level of accuracy discussed above also when scales are varied. In Fig. 4, we show the effect of varying the renormalisation scale $\mu _\textrm{R}$ (red bands) and the factorisation scale $\mu _\textrm{F}$ (blue bands) by a factor of 2 up and down with respect to Q relative to the central-scale choice $\mu _\textrm{R}=\mu _\textrm{F}=Q$. The left plot has been obtained with $Q=2$ GeV while the right plot with $Q=10$ GeV. In each of them the top panel shows variations at NLO, the central panel at NNLO, and the bottom panel at N$^3$LO. We did not include the LO panel because at this order inclusive DIS structure functions are independent of $\mu _\textrm{R}$ while $\mu _\textrm{F}$ gives rise to very large bands.¹¹ As expected, scale-variation bands shrink significantly moving from NLO to N$^3$LO at both scales. However, the reduction is much more pronounced at $Q=10$ GeV than at $Q=2$ GeV, as a consequence of the decrease in $\alpha _s$ value.

5 Conclusion

As N$^3$LO PDFs start to emerge, see e.g. Refs. [89, 90], it will become increasingly important to have reliable N$^3$LO predictions for inclusive DIS cross sections available. Indeed, DIS measurements are and will likely remain one of the main sources of experimental information that enter modern determinations of PDFs. Moreover DIS is one of the very few processes for which N$^3$LO corrections to the partonic cross sections, albeit only in the quark massless limit, are exactly known.

In this paper, we have benchmarked the implementation of the massless DIS structure functions to N$^3$LO accuracy by comparing the predictions provided by two widely used codes: apfel++ [28, 29] and hoppet [30]. In this benchmark, we considered both NC and CC structure functions relevant to the computation of DIS cross sections respectively characterised by the exchange of a neutral ($\gamma /Z$) and a charged ($W^{\pm }$) virtual vector boson. The numerical setup closely follows that of Ref. [36]. Specifically, we used a realistic set of initial-scale PDFs that were evolved to the relevant scales before convoluting them with the appropriate DIS coefficient functions. This workflow was independently implemented in both apfel++ and hoppet before comparing the respective predictions. On top of the single structure functions, we also compared reduced cross sections as those delivered by the HERA experiments [7].

We found a relative agreement between apfel++ and hoppet at the $10^{-5}$ level or better over a very wide kinematic range in $Q\in [2:100]$ GeV and $x_\textrm{B}\in [10^{-5}:0.9]$ for all structure functions and reduced cross sections. Additionally, we also investigated the perturbative convergence by comparing predictions for $F_{2}^\textrm{NC}$ at all available perturbative orders, i.e. from LO to N$^3$LO, and by estimating the effect of renormalisation and factorisation scale variations. We found the expected pattern according to which $F_{2}^\textrm{NC}$ exhibits a good perturbative convergence, especially for large values of Q. apfel++ and hoppet were benchmarked also against scale variations.

The benchmark carried out in this paper, thanks to its high accuracy level, provides a solid reference for any future implementation of the DIS coefficient functions up to N$^3$LO perturbative accuracy. In order to make our results fully reproducible and facilitate the comparison to future implementations, we made the code used for this benchmark available at:

https://github.com/alexanderkarlberg/n3lo-structure-function-benchmarks,

where we also provide a short documentation and the suite of Matplotlib scripts that we used to produce the plots shown in this paper.

Appendices

Appendix A: Comments on the $y\rightarrow 1$ behaviour of the coefficient functions

In the course of this benchmark, we have encountered some minor differences between the exact N$^3$LO coefficient functions and the parameterisations used here and given in Refs. [19,20,21, 23]. The differences arise for $y \gtrsim 0.9$ in the regular part of the non-singlet coefficient functions,¹² Although the difference is phenomenologically negligible, it does have a small impact in precision studies like the benchmark presented here. The largest relative difference is found in the $c^{(3)}_{2,q}$ coefficient function, on which we focus our attention.

Fig. 5

Left: the regular $\mathcal {O}(\alpha _s ^3)$ coefficient function $c^{(3)}_{2,\textrm{q, reg}}$ as a function of $1-y$ plotted as a ratio to the exact expression. In green we show the parametrisation of Ref. [20]. The dashed lines show the large-y expansion including progressively more terms. Right: the same plot but now showing 1 minus the ratio on a log-scale to highlight the relative agreement

In general, a DIS coefficient function C receives three different contributions:

• The singular piece $[C_\textrm{sing}(y)]_+$, which is a combination of terms of the kind $\left[ \frac{\ln ^i(1-y)}{1-y}\right] _{+}$. The $+$-prescription is defined as:

  $$\begin{aligned} \int _0^1 dy [C_\textrm{sing}(y)]_+ f(y){} & {} = \int _0^1 dy \, C_\textrm{sing}(y) [f(y) \nonumber \\ {}{} & {} \quad - f(1)], \end{aligned}$$

  (A.1)

  and has the effect of regularising the otherwise non-integrable singularities at $y=1$.

• The regular piece $C_\textrm{reg}(y)$, which in general can be very complicated but in the $y\rightarrow 1$ limit develops integrable singular terms of the kind $\ln ^i(1-y)$.

• The local piece $C_\textrm{loc}\delta (1-y)$, where $C_\textrm{loc}$ is a numerical constant.

Although the singular piece $[C_\textrm{sing}(y)]_+$ dominates the coefficient function for $y\rightarrow 1$, this is not the case when it is convoluted with a parton distribution as in Eq. (8), since the $+$-prescription effectively generates a factor of $1-y$. In the limit $y\rightarrow 1$, this can be seen schematically as follows:

$$\begin{aligned} C\otimes f&= \int _{x_\textrm{B}}^1 \frac{dy}{y}\left\{ C_{\textrm{reg}}(y) \right. \nonumber \\ {}&\quad \left. + \left[ C_{\textrm{sing}}(y)\right] _++C_\textrm{loc}\delta (1-y)\right\} f\left( \frac{x_\textrm{B}}{y}\right) \nonumber \\&\simeq \int _{x_\textrm{B}}^1 dy \left\{ C_{\textrm{reg}}(y)f(x_\textrm{B}) + (1-y)C_{\textrm{sing}}(y)\right. \nonumber \\&\quad \times \left. \left[ f(x_\textrm{B})+x_\textrm{B} f'(x_\textrm{B})\right] + \dots \right\} \,, \end{aligned}$$

(A.2)

where in the second line we have expanded $f(x_\textrm{B}/y)/y$ around $y=1$ and neglected the local piece as well as the additional terms proportional to $\ln ^{i+1}(1-x_\textrm{B})\delta (1-y)$ generated by the $+$-prescription. Therefore, in order to achieve accurate results when parameterising C, it is necessary to correctly account for the large-y behaviour of $C_\textrm{reg}$.

The solid green curve on the l.h.s. of Fig. 5 shows the ratio between the regular part of the parametrisation for $c_{2,q}^{(3)}$ given in Eq. (4.11) of Ref. [20] and its exact counterpart as a function of $1-y$. As can be seen, the ratio increases as $1-y$ approaches zero and, in the range shown on the plot, it reaches $8\%$.

In order to investigate this difference, we considered the large-y limit of the regular part of all $\mathcal {O}(\alpha _s^3)$ non-singlet coefficient functions, which in this region admit the following expansion:

$$\begin{aligned} c_{k,q,\textrm{reg}}^{(3)}(y) \simeq \sum _{i=1}^5 L_{i}^{(k)} \ln ^i(1-y)\,, \end{aligned}$$

(A.3)

with $k=2,3,L$. The coefficients $L_i$ can be found in Refs. [20, 23]. Since these coefficients are non-trivial to derive, we recomputed them finding two minor typos in the expressions for $c^{(3)}_{L,q,\textrm{reg}}$ reported in Ref. [20]. Specifically, for the coefficients of $c^{(3)}_{2,q,\textrm{reg}}$ we find:

$$\begin{aligned} L_5^{(2)}= & {} - 8 {C^{\, 3}_F}, \end{aligned}$$

(A.4)

$$\begin{aligned} L_4^{(2)}= & {} {220 \over 9} {C^{}_A}{C^{\, 2}_F}+ 92 {C^{\, 3}_F}- {40 \over 9} {C^{\, 2}_F}\, {n^{}_{\! f}}, \end{aligned}$$

(A.5)

$$\begin{aligned} L_3^{(2)}= & {} - {484 \over 27} {C^{\, 2}_A}{C^{}_F}- {C^{}_A}{C^{\, 2}_F}\Bigg [ {10976 \over 27} - 64\, {\zeta _{\,\! 2}} \Bigg ]\nonumber \\{} & {} - {C^{\, 3}_F}[ 38 - 32\, {\zeta _{\,\! 2}}]\nonumber \\{} & {} + {176 \over 27} {C^{}_A}{C^{}_F}\, {n^{}_{\! f}}+ {1832 \over 27} {C^{\, 2}_F}\, {n^{}_{\! f}}- {16 \over 27} {C^{}_F}\, {n^{\,2}_{\! f}}, \end{aligned}$$

(A.6)

$$\begin{aligned} L_2^{(2)}= & {} {C^{\, 2}_A}{C^{}_F}\Bigg [ {11408 \over 27} - {266 \over 3} {\zeta _{\,\! 2}} - 32\, {\zeta _{\,\! 3}} \Bigg ]\nonumber \\{} & {} + {C^{}_A}{C^{\, 2}_F}\Bigg [ {11501 \over 9} - 292\, {\zeta _{\,\! 2}} - 160\, {\zeta _{\,\! 3}} \Bigg ] \nonumber \\{} & {} - {C^{\, 3}_F}\Bigg [ {1199 \over 3} + 688\, {\zeta _{\,\! 2}} + 48\, {\zeta _{\,\! 3}} \Bigg ]\nonumber \\{} & {} - {C^{}_A}{C^{}_F}\, {n^{}_{\! f}}\Bigg [ {3694 \over 27} - {64 \over 3} {\zeta _{\,\! 2}} \Bigg ]\nonumber \\{} & {} - {C^{\, 2}_F}\, {n^{}_{\! f}}\Bigg [ {2006 \over 9} - {16 \over 3}\, {\zeta _{\,\! 2}} \Bigg ] + {296 \over 27} {C^{}_F}\, {n^{\,2}_{\! f}}, \end{aligned}$$

(A.7)

$$\begin{aligned} L_1^{(2)}= & {} - {C^{\, 2}_A}{C^{}_F}\Bigg [ {215866 \over 81} - 824 {\zeta _{\,\! 2}} - {1696 \over 3}\, {\zeta _{\,\! 3}} + {304 \over 5} {\zeta _{2}^{\,2}}\Bigg ]\nonumber \\{} & {} + {C^{}_A}{C^{\, 2}_F}\Bigg [ {126559 \over 162} + 872 {\zeta _{\,\! 2}} + 792 {\zeta _{\,\! 3}} - {1916 \over 5} {\zeta _{2}^{\,2}}\Bigg ]\nonumber \\{} & {} + {C^{\, 3}_F}\Bigg [ {157 \over 6} + {1268 \over 3} {\zeta _{\,\! 2}} - 272\, {\zeta _{\,\! 3}} + 488 {\zeta _{2}^{\,2}}\Bigg ] \nonumber \\{} & {} + {C^{}_A}{C^{}_F}\, {n^{}_{\! f}}\Bigg [ {64580 \over 81} - {1292 \over 9}\, {\zeta _{\,\! 2}} - {304 \over 3}\, {\zeta _{\,\! 3}} \Bigg ]\nonumber \\{} & {} - {C^{\, 2}_F}\, {n^{}_{\! f}}\Bigg [ {4445 \over 81} + 208 {\zeta _{\,\! 2}} - {208 \over 3} {\zeta _{\,\! 3}} \Bigg ]\nonumber \\{} & {} - {C^{}_F}\, {n^{\,2}_{\! f}}\Bigg [ {4432 \over 81} - {32 \over 9} {\zeta _{\,\! 2}} \Bigg ], \end{aligned}$$

(A.8)

which agree with the results of Ref. [23]. Similarly, for the coefficients of $c^{(3)}_{3,q, \textrm{reg}}$ we have:

$$\begin{aligned} L_5^{(3)}= & {} - 8 {C^{\, 3}_F}, \end{aligned}$$

(A.9)

$$\begin{aligned} L_4^{(3)}= & {} {220 \over 9} {C^{}_A}{C^{\, 2}_F}+ 84 {C^{\, 3}_F}- {40 \over 9} {C^{\, 2}_F}\, {n^{}_{\! f}}, \end{aligned}$$

(A.10)

$$\begin{aligned} L_3^{(3)}= & {} - {484 \over 27} {C^{\, 2}_A}{C^{}_F}\nonumber \\{} & {} - {C^{}_A}{C^{\, 2}_F}\Bigg [ {9056 \over 27} - 32\, {\zeta _{\,\! 2}} \Bigg ] - {C^{\, 3}_F}[ 110 - 96\, {\zeta _{\,\! 2}}]\nonumber \\{} & {} + {176 \over 27} {C^{}_A}{C^{}_F}\, {n^{}_{\! f}}+ {1640 \over 27} {C^{\, 2}_F}\, {n^{}_{\! f}}- {16 \over 27} {C^{}_F}\, {n^{\,2}_{\! f}}, \end{aligned}$$

(A.11)

$$\begin{aligned} L_2^{(3)}= & {} {C^{\, 2}_A}{C^{}_F}\Bigg [ {7580 \over 27} - {98 \over 3} {\zeta _{\,\! 2}} \Bigg ]\nonumber \\{} & {} + {C^{}_A}{C^{\, 2}_F}\Bigg [ {12031 \over 9} - 372\, {\zeta _{\,\! 2}} - 240\, {\zeta _{\,\! 3}} \Bigg ]\nonumber \\{} & {} - {C^{\, 3}_F}\Bigg [ {1097 \over 3} + 656\, {\zeta _{\,\! 2}} + 16\, {\zeta _{\,\! 3}} \Bigg ]\nonumber \\{} & {} - {C^{}_A}{C^{}_F}\, {n^{}_{\! f}}\Bigg [ {2734 \over 27} - {16 \over 3} {\zeta _{\,\! 2}} \Bigg ]\nonumber \\{} & {} - {C^{\, 2}_F}\, {n^{}_{\! f}}\Bigg [ {2098 \over 9} - {112 \over 3}\, {\zeta _{\,\! 2}} \Bigg ] + {248 \over 27} {C^{}_F}\, {n^{\,2}_{\! f}}, \end{aligned}$$

(A.12)

$$\begin{aligned} L_1^{(3)}= & {} - {C^{\, 2}_A}{C^{}_F}\Bigg [ {138598 \over 81} - {4408 \over 9} {\zeta _{\,\! 2}} - 272\, {\zeta _{\,\! 3}} + {176 \over 5} {\zeta _{2}^{\,2}}\Bigg ]\nonumber \\{} & {} - {C^{}_A}{C^{\, 2}_F}\Bigg [ {69833 \over 162} - {12568 \over 9} {\zeta _{\,\! 2}} - {1904 \over 3} {\zeta _{\,\! 3}} + {764 \over 5} {\zeta _{2}^{\,2}}\Bigg ]\nonumber \\{} & {} + {C^{\, 3}_F}\Bigg [ {1741 \over 6} + {1220 \over 3} {\zeta _{\,\! 2}} + 480\, {\zeta _{\,\! 3}} - {376 \over 5} {\zeta _{2}^{\,2}}\Bigg ]\nonumber \\{} & {} + {C^{}_A}{C^{}_F}\, {n^{}_{\! f}}\Bigg [ {45260 \over 81} - 108\, {\zeta _{\,\! 2}} - 16\, {\zeta _{\,\! 3}} \Bigg ] \nonumber \\{} & {} + {C^{\, 2}_F}\, {n^{}_{\! f}}\Bigg [ {9763 \over 81} - {2224 \over 9} {\zeta _{\,\! 2}} - {112 \over 3} {\zeta _{\,\! 3}} \Bigg ]\nonumber \\{} & {} - {C^{}_F}\, {n^{\,2}_{\! f}}\Bigg [ {3520 \over 81} - {32 \over 9} {\zeta _{\,\! 2}} \Bigg ], \end{aligned}$$

(A.13)

Table 2 NLO stucture functions with NLL evolution at $Q = 2$ GeV

which agree with the results in Ref. [23]. Finally, for the coefficients of $c^{(3)}_{L,q, \textrm{reg}}$ we find:

$$\begin{aligned} L_4^{(L)}\!= & {} 8 C_F^{\,3}, \end{aligned}$$

(A.14)

$$\begin{aligned} L_3^{(L)}\!= & {} C_A C_F^{\,2} \left[ - \frac{640}{9} + 32{\zeta _{\,\! 2}} \right] \nonumber \\{} & {} + C_F^{\,3} \Big [ 72 - 64{\zeta _{\,\! 2}} \Big ] {+ \frac{64}{9} C_F^2 {n^{}_{\! f}}}, \end{aligned}$$

(A.15)

$$\begin{aligned} L_2^{(L)}\!= & {} C_A^{\,2} C_F \,\left[ \frac{1276}{9} - 56{\zeta _{\,\! 2}} - 32{\zeta _{\,\! 3}} \right] \nonumber \\{} & {} + C_A C_F^{\,2} \,\left[ -\frac{530}{9} + 80{\zeta _{\,\! 2}} + 80{\zeta _{\,\! 3}} \right] \nonumber \\{} & {} + C_F^{\,3} \,\Big [ -34 - 32{\zeta _{\,\! 2}} -32{\zeta _{\,\! 3}} \Big ]\nonumber \\{} & {} + C_A C_F {n^{}_{\! f}}\,\left[ -\frac{320}{9} + 16{\zeta _{\,\! 2}} \right] \nonumber \\{} & {} + C_F^{\,2} {n^{}_{\! f}}\,\left[ \frac{92}{9} - 32{\zeta _{\,\! 2}} \right] + \frac{16}{9} C_F {n^{\,2}_{\! f}}, \end{aligned}$$

(A.16)

$$\begin{aligned} L_1^{(L)}\!= & {} C_A^{\,2} C_F \,\left[ - \frac{25756}{27} + \frac{3008}{9}{\zeta _{\,\! 2}} + \frac{880}{3}{\zeta _{\,\! 3}} - \frac{128}{5}{\zeta _{2}^{\,2}}\right] \nonumber \\{} & {} + C_A C_F^{\,2} \,\left[ \frac{32732}{27} - \frac{4720}{9}{\zeta _{\,\! 2}}\right. \nonumber \\{} & {} \left. + \frac{472}{3}{\zeta _{\,\! 3}} - \frac{1152}{5}{\zeta _{2}^{\,2}}\right] + C_F^{\,3} \,\Bigg [ - 264 \nonumber \\{} & {} \left. + 16{\zeta _{\,\! 2}} - 752{\zeta _{\,\! 3}} {+ \frac{2816}{5}{\zeta _{2}^{\,2}}} \right] \nonumber \\{} & {} + C_A C_F {n^{}_{\! f}}\,\left[ \frac{6640}{27} - \frac{320}{9}{\zeta _{\,\! 2}} - \frac{256}{3}{\zeta _{\,\! 3}} \right] \nonumber \\{} & {} + C_F^{\,2} {n^{}_{\! f}}\,\left[ -\frac{4736}{27} + \frac{352}{9}{\zeta _{\,\! 2}} + \frac{320}{3}{\zeta _{\,\! 3}} \right] \nonumber \\{} & {} - \frac{304}{27} C_F {n^{\,2}_{\! f}}, \end{aligned}$$

(A.17)

which also agree with the expressions given in Ref. [20], except for the two terms highlighted in blue.

Table 3 NLO stucture functions with NLL evolution at $Q = 50$ GeV

Table 4 NLO stucture functions with NLL evolution at $Q = 100$ GeV

Table 5 NNLO stucture functions with NNLL evolution at $Q = 2$ GeV

Table 6 NNLO stucture functions with NNLL evolution at $Q = 50$ GeV

Table 7 NNLO stucture functions with NNLL evolution at $Q = 100$ GeV

Table 8 N$^{3}$LO stucture functions with NNLL evolution at $Q = 2$ GeV

Table 9 N$^{3}$LO stucture functions with NNLL evolution at $Q = 50$ GeV

Table 10 N$^{3}$LO stucture functions with NNLL evolution at $Q = 100$ GeV

On the l.h.s. plot of Fig. 5, we also show the large-y expansion of Eqs. (A.4)–(A.8) retaining progressively more terms. It can be seen that the agreement between the exact coefficient function and the large-y expansion improves as y increases, as expected. We conclude that the parameterisation for the regular part of $c_{2,q}^{(3)}$ reported in Eq. (4.11) of Ref. [20] does not fully account for its large-y behaviour, but we also stress that the phenomenological impact is negligible.

On the r.h.s. of Fig. 5, we also show 1 minus the ratio to the exact expression to highlight the relative agreement between the various curves. From this plot, we notice an apparent degradation of the overall precision of the exact expression as provided in Ref. [20] as y approaches one (see the oscillations of the red curve).¹³ In hoppet, where the exact expressions have been implemented, this causes some issues as the numerical convolution requires the evaluation of the coefficient functions at values of y that in double precision are indistinguishable from 1. For this reason, we switch to the large-y expressions close to $y=1$.

As stated above, the difference between using exact and parametrised coefficient functions is phenomenologically negligible at the level of the structure functions. Indeed, if we reproduce the benchmark Tables 5, 6, 7, 8, 9, and 10 using the exact N$^3$LO expressions in hoppet (but keeping the parametrisations at NNLO), they typically differ from the tables obtained with the parameterised expressions only in the last (fifth) digit, and often not at all. Our benchmark program, StructureFunctionsJoint.cc, can be modified to use the exact expressions in hoppet by setting the flag param_coefs to false. Finally, we provide a Fortran file, c_ns_reg_large_x.f, with the large-y expressions given above.

Appendix B: Benchmark tables

In this appendix, we collect benchmark tables for all of the inclusive DIS structure functions. Results are presented for both NC and CC channels at NLO, NNLO, and N$^3$LO accuracy, for $Q=2, 50, 100$ GeV, and at values of $x_\textrm{B}$ ranging from $10^{-5}$ to 0.9. Details on the numerical setup can be found in Sect. 3. The numbers reported in Tables 2, 3, 4, 5, 6, 7, 8, 9, and 10 agree between apfel++ and hoppet within the digits shown. The code that produces the tables can be found in StructureFunctionsJoint.cc.