Soft corrections to inclusive DIS at four loops and beyond

TL;DR

This paper advances the understanding of threshold corrections in deep-inelastic scattering by calculating four-loop contributions to soft-gluon exponentiation, showing these corrections are small and improve the stability of perturbative QCD predictions.

Contribution

It derives the fourth-order contribution to the coefficient of the quark form factor and the N$^4$LL exponentiation coefficient in DIS, extending soft-gluon resummation to five loops.

Findings

N$^4$LL corrections are small.

The results stabilize the perturbative expansion.

An approximation scheme yields accurate N$^4$LL results for full QCD.

Abstract

We study the threshold corrections to the structure functions in deep-inelastic scattering (DIS) at the fifth logarithmic (N$^4$LL) order of the soft-gluon exponentiation in massless perturbative QCD. Using recent results for the splitting functions and the quark form factor, we derive the fourth-order contribution to the coefficient $f^{\rm q}$ of the form factor and from it the N$^4$LL part of the exponentiation coefficient $B^{\rm DIS}$ in the limit of a large number of colours. An approximation scheme is shown that leads to sufficiently accurate N$^4$LL results for full QCD. The N$^4$LL corrections are small and lead to a further stabilization of the perturbative expansion for the soft-gluon exponent.

Equations

$$C^N(Q^2)=g_0(Q^2)\cdot \exp [G^N(Q^2)]+\mathcal{O}(N^{-1}{\ln }^{n}N),$$

$$\displaystyle G^{\,N}=\int_{0}^{1}dz\>\frac{z^{\,N-1}-1}{1-z}~{}\bigg{[}\int_{\mu_{f}^{2}}^{(1-z)Q^{2}}\frac{dq^{2}}{q^{2}}\,A^{\rm q}\big{(}\alpha_{\rm s}(q^{2})\big{)}+B^{\>\!\rm DIS}\big{(}\alpha_{\rm s}((1-z)q^{2})\big{)}\bigg{]}\>.$$

$$G^N$$

$$g_0=1+a_{\mathrm{s}}{g}_{01}+a_{\mathrm{s}}^2{g}_{02}+a_{\mathrm{s}}^3{g}_{03}+\mathcal{O}(a_{\mathrm{s}}^4).$$

$$G_4^{\mathrm{q}}=2B_4^{\mathrm{q}}+f_4^{\mathrm{q}}+{\beta }_2{f}_{01}^{\mathrm{q}}+{\beta }_1{f}_{02}^{\mathrm{q}}+{\beta }_0{f}_{03}^{\mathrm{q}}+\mathcal{O}(ϵ),$$

$$\displaystyle f_{4}^{\,\rm q}\Bigl{|}_{{\rm L}{n_{c}}}$$

$$B_4^{\mathrm{DIS}}$$

$$\displaystyle\;B_{4}^{\,\rm DIS}\Bigl{|}_{{\rm L}{n_{c}}}\!\!$$

Full text

1 Introduction

The Wilson coefficients (coefficient functions) for the structure functions of inclusive DIS have been a subject of research since the early days of QCD. These quantities are not only relevant for determining the parton distribution functions (PDFs) and the strong coupling constant $\alpha_{\rm s}$ using structure function data, see, e.g., [1], but also to other processes and less inclusive observables in DIS, see, e.g., [2]. The main Wilson coefficients for DIS are presently known up to the third order in $\alpha_{\rm s}$ in massless perturbative QCD [3]. Their perturbative expansion is well-behaved except close to the kinematic endpoints $x=0$ and $x=1$ of the Bjorken variable. The dominant terms $\ln^{\,\ell}(1-x)/(1-x)_{+}$ in the latter (threshold) limit are resummed by the soft-gluon exponentiation, see, e.g., [4, 5, 6], which is best formulated in Mellin $N$-space [4]. So far this resummation has been performed up to the next-to-next-to-next-to-leading logarithmic (N3LL) accuracy [7].

The threshold resummation coefficients are closely related to the large-$x$ limit of the quark-quark splitting functions for the PDFs and to the quark form factor [8, 9] which are both fully known to order ${\alpha_{\rm s}^{\>\!3}}$ [10, 11, 12, 13]. Recently the computations of these quantities have been extended to order ${\alpha_{\rm s}^{\>\!4}}$ in the (L$n_{c}$) limit of a large number of colours [14, 15]. Together with approximate results for the ${n_{\!f}}$-independent [16] and exact expression for the ${n_{\!f}}$-dependent contributions to the cusp anomalous dimension in full QCD [17] these results facilitate the effective extension of the threshold resummation for the DIS Wilson coefficients to the next (N4LL) logarithmic order. In the following we recall the theoretical framework, present the N4LL resummation coefficient and briefly address the numerical implications of this result for the resummation of DIS in QCD.

2 Theoretical framework and new fourth-order coefficients

The all-order large-$N$ behaviour of the DIS Wilson coefficients for $F_{1}$, $F_{2}$ and $F_{3}$ can be written as

[TABLE]

where the resummation exponent $G^{\,N}$ of the dominant $N^{0}\ln^{\>\!n}N$ contributions is given by [18]

[TABLE]

Here $A^{\rm q}$ is the (light-like) quark cusp anomalous dimension and $B^{\>\!\rm DIS}$ is the resummation coefficient for DIS. Both have perturbative series, $A^{\rm q}=\sum_{i}\,a_{s}^{i}\,A^{\rm q}_{i}$ etc, in terms of the strong coupling which we normalize as $a_{\rm s}\equiv\alpha_{\rm s}/4\pi$. Performing the integrations one can organize the exponent as

[TABLE]

where $\lambda=\beta_{0}\>\!a_{\rm s}\ln\,N$ or $\lambda=\beta_{0}\>\!a_{\rm s}\ln\,\widetilde{N}$ with $\ln\,\widetilde{N}=\ln\,N+\gamma_{\,\rm e\,}$. The first $n\!+\!1$ terms in (2.3) are required for the resummation at NnLL accuracy. The N2LL and N3LL contributions to $G^{\,N}$ have been derived in [19, 7]; explicit expressions can be found in (3.3) – (3.6) of [7]. The lengthy new function $g^{(5)}(\lambda)$ entering at N4LL will be presented in [20]. The $N$-independent prefactor $g_{0}$ is presently known to order ${\alpha_{\rm s}^{\>\!3}}$ from the all-$N$ calculation in [3], see (4.6) – (4.8) of [7],

[TABLE]

The resummation to N4LL requires the terms up to $A_{5}^{\rm q}$ and $B_{4}^{\>\!\rm DIS}$ in their corresponding expansions. The impact of the former quantity, for which a first estimate has been obtained in [21], is very small. $B^{\>\!\rm DIS}$ can be calculated from knowledge of the quark form factor or the DIS Wilson coefficients. The form factor satisfies a differential equation which follows from the renormalization group and gauge invariance. Its solution can be found in terms of the cusp anomalous dimension $A^{\rm q}$ and the function $G^{\rm q}$ containing the quantity $f^{\,\rm q}$ related to a universal eikonal anomalous dimension and the coefficient $B^{\>\!\rm q}$ of $\delta(1-x)$ in the quark-quark splitting function. The four-loop coefficient of $G^{\>\!\rm q}$ (which appears in the $1/\epsilon$ coefficient in the solution of the form factor) can be written as

[TABLE]

where the quantities $f_{0{\rm n}}^{\,\rm q}$ are (combinations of) known lower-order coefficients of $G^{\,\rm q}$, see [12] and (20) of [9]. Hence $f_{4}^{\,\rm q}$ can be determined in the large-$n_{c}$ limit from the results of [14, 15]. We find

[TABLE]

The ${a_{\rm s}^{\>\!4}}$ contribution to resummation $B^{\,\rm DIS}$ reads, in terms of the genuine four-loop contributions $f_{4}^{\,\rm q}$ and $\,B_{4}^{\,\rm q}$, which are exactly known only in the L$n_{c}$ limit for now, and lower-order coefficients,

[TABLE]

where $g_{0i}$ are to be taken without the $\gamma_{\,\sf e\,}$ terms in (4.6) – (4.8) of [7]. Its explicit form is given by

[TABLE]

3 Numerical implications

The lower-order coefficients $B_{l}^{\,\rm DIS}$ have the same structure as (2.7), i.e., they contain $-f_{l}^{\,\rm q}-B_{l}^{\,\rm q}$ and lower-order coefficients. Therefore, by comparing the exact results to an approximation at NlLL in which the L$n_{c}$ expression for $-f_{l}^{\,\rm q}-B_{l}^{\,\rm q}$ is used together with the exact lower-order coefficients, we can check whether (2.7) with the L$n_{c}$ results for $f_{4}^{\,\rm q}$ and $B_{4}^{\,\rm q}$ can be expected to provide a good approximation for $B_{4}^{\>\!\rm DIS}$ and hence $G^{\,N}$ at the N4LL accuracy of full QCD.

This comparison is carried out in Fig. 1 for $l=2$ and $l=3$ (at $l=1$ there is no difference between the L$n_{c}$ limit and full QCD). The L$n_{c}$ curves are off by less than 0.5% at N2LL and 0.25% at N3LL for $G^{\,\rm DIS}$ in the $N$-range shown and, at $x\leq 0.9$, for the convolution of its exponential with a schematic but sufficiently realistic form for a quark PDF. Therefore we can safely expect that the L$n_{c}$ numbers will deviate from (presumably exceed) the exact QCD results by well below 1%.

The cumulative effect, relative to the NLL results, of the exact N2LL and N3LL contributions and our new N4LL corrections, as above determined using the L$n_{c}$ limit of $-f_{l}^{\,\rm q}-B_{l}^{\,\rm q}$ in (2.7), is illustrated in Fig. 2. Unlike the N3LL contribution, the N4LL correction is almost negligible at $N\leq 15$ and $x\leq 0.9$. Even at $N=40$, the functions $g^{(n)}(\lambda)$ add only 6%, 1.6% and 1% to the NLL result, respectively, for $n=2$, $n=3$ and $n=4$, where the latter L$n_{c}$ result is presumably a slight overestimate. The corresponding N2LL, N3LL and N4LL percentages for the convolution of $\exp G^{\,N}$ with $xf=x^{0.5}(1-x)^{3}$ at $x=0.95$ read 9.5%, 1.5% and 0.5%, where we have performed the Mellin inversion using a standard contour, see, e.g., [22], which constitutes a ‘minimal prescription’ contour [4] in the context of the present exponentiation. It appears that the expansion of $G^{\,N}$ to N4LL for the structure functions in inclusive DIS is sufficient for all practical purposes.

4 Summary and outlook

We have studied the soft-gluon exponentiation (SGE) of inclusive DIS at the fifth logarithmic (N4LL) order. Recent four-loop results on splitting functions and the quark form factor [14, 15] facilitate the exact determination of the form-factor coefficient $f^{\,\rm q}$ and the SGE coefficient $B^{\,\rm DIS}$ at order ${\alpha_{\rm s}^{\>\!4}}$ in the large-$n_{c}$ (L$n_{c}$) limit. Both coefficients are relevant beyond the context of DIS: Like the lightlike quark and gluon cusp anomalous dimensions $A^{\rm q,g}$ [10], the quantities $f^{\,\rm q,g}$ are maximally non-Abelian and related by a simple Casimir scaling up to three loops. We expect that the generalized Casimir scaling of [16] also applies to $f^{\,\rm q,g}$, hence our result (2.6) fixes also $f^{\,\rm g}$ at large $n_{c}$. The coefficient here called $B^{\,\rm DIS}$ is due to the outgoing unobserved quark; hence it contributes to the SGE for many other processes including, e.g., direct photon production [6].

The L$n_{c}$ approximation to the N4LL resummation exponent $G^{\,N}$ for inclusive DIS, defined as discussed above, is sufficiently accurate to demonstrate that the N4LL corrections are small: they contribute well below 1% over a wide range in $N$ and $x$. As shown in [23], the $1/N\ln^{\>\!\ell}N$ non-SGE contributions are larger; the highest four of these logarithms are currently known to all orders [23, 24] – recall the parameter $\xi_{\rm DIS_{4}}$ unspecified in [23] was fixed in [24]. We have considered the case of ${n_{\!f}}=3$ light flavours. In electromagnetic and neutral-current DIS, also charm production close to threshold needs to be taken into account beyond the threshold for $c\bar{c}$ production, see [25].