Four-loop QCD propagators and vertices with one vanishing external momentum

We have computed the self-energies and a set of three-particle vertex functions for massless QCD at the four-loop level in the MSbar renormalization scheme. The vertex functions are evaluated at points where one of the momenta vanishes. Analytical results are obtained for a generic gauge group and with the full gauge dependence, which was made possible by extensive use of the Forcer program for massless four-loop propagator integrals. The bare results in dimensional regularization are provided in terms of master integrals and rational coefficients; the latter are exact in any space-time dimension. Our results can be used for further precision investigations of the perturbative behaviour of the theory in schemes other than MSbar. As an example, we derive the five-loop beta function in a relatively common alternative, the minimal momentum subtraction (MiniMOM) scheme.

The main aim of this work is to provide the self-energies and a set of vertices with one vanishing external momentum for massless QCD at the four-loop level. The unrenormalized results are exact in terms of ε = (D − 4)/2, where D is the space-time dimension, and four-loop master integrals [35,36]. The renormalized D = 4 results are given in the MS scheme. The computation has been performed for a general gauge group and in an arbitrary covariant linear gauge, by using the Forcer program [37][38][39] for massless four-loop propagator-type integrals. For the vertices, setting one of the momenta to zero effectively reduces vertex integrals to propagator-type integrals. In QCD this does not create IR divergences, which means the poles do not change. At the three-loop level, similar computations were performed in ref. [40], but with an expansion in ε. In addition, studies of QCD vertices in perturbation theory for various configurations include refs. [41][42][43][44][45][46][47][48][49][50][51][52][53].
We compute all QCD vertices in a general linear covariant gauge, with the exception of the four-gluon vertex for which there are at least three difficulties: first, two momenta have to be nullified before the diagrams become propagator-like. Second, the number of diagrams is large at four loops. Third, the colour structure for a generic group is no longer an overall factor, but will be term dependent. Additionally, the renormalization constant is completely determined through the Slavnov-Taylor identities [54,55], which means the quartic gluon vertex does not provide extra information in this context.
A direct application of our results is to compute conversion factors from the MS scheme to momentum subtraction schemes, see, e.g., refs. [41,44], for renormalization group functions. Unlike the MS scheme, momentum subtraction schemes are defined in a regularization-independent way. In these schemes, the field renormalizations are performed such that finite radiative corrections on propagators are absorbed as well as divergences and hence they coincide with their tree-level values at the renormalization point. Then one of (or an arbitrary linear combination of) the vertex functions is normalized to its tree-level value and the other vertices are fixed via the Slavnov-Taylor identities. Common choices for the subtraction point of the vertex are a symmetric point (referred as MOM schemes) and an asymmetric point where one of the momenta is nullified, sometimes referred as MOM schemes. The latter choice corresponds to our result for the vertex functions. Indeed, ref. [40] derived four-loop beta functions in four particular MOM schemes from that in the MS scheme by computing conversion factors via finite parts of two-and three-point functions in the MS scheme.
As an example application, we provide the five-loop beta function in the minimal momentum subtraction (MiniMOM) scheme introduced in ref. [56], thus extending previous results [56,57] by one order in the coupling constant. This scheme, see the preceeding references for a detailed discussion, is more convenient than MS for extending analyses of the strong coupling constant and its scale dependence into the non-perturbative regime, e.g., via lattice QCD; for a recent analysis see ref. [58]. In the perturbative regime the MiniMOM scheme provides an alternative to MS for studying the behaviour and truncation uncertainty of the perturbation series for benchmark quantities such as the R-ratio in e + e − annihilation and the Higgs-boson decay to gluons, see refs. [59][60][61].
The remainder of this article is organized as follows: In section 2 we specify our notations for the self-energies and vertex functions and recall their renormalization. In section 3 we address technical details and checks of the computation. Due to their length, the results for the renormalized self-energies and vertex functions for a generic gauge group and a general linear covariant gauge are deferred to appendix A. In section 4 the five-loop beta function is presented in the MiniMOM scheme for QCD in the Landau gauge; the general result and the corresponding MS-to-MiniMOM conversion factor for the coupling constant are provided in appendix B. In section 5 we summarize and give a brief outlook.
Our result can be obtained in a digital form as ancillary files of this article on the preprint server https://arXiv.org. They are also available from the authors upon request. The files contain the bare results for the self-energies and vertices in terms of master integrals with coefficients that are exact for any dimension D, as well as the results in the MS scheme for D = 4. The notations in these files can be found in appendix C.

Preliminaries
We first summarize the notations for self-energies and vertex functions with one vanishing momentum presented in this article. In most cases we follow the conventions in ref. [40]. 1

Self energies
The gluon, ghost and quark self-energies (figure 1) are of the form Π ab µν (q) = −δ ab (q 2 g µν − q µ q ν )Π(q 2 ), (2.1) Π ab (q) = δ ab q 2Π (q 2 ), (2.2) Σ ij (q) = δ ij / qΣ V (q 2 ). (2.3) The colour indices are understood such that a and b are for the adjoint representation of the gauge group, i and j for the representation to which the quark belongs. The form factors Π(q 2 ),Π(q 2 ) and Σ V (q 2 ) can easily be extracted from contributions of the corresponding one-particle irreducible diagrams by applying projection operators [40] (the same holds for the vertex functions discussed below). They are related to the full gluon, ghost and quark propagators as follows: ∆ ab (q) = δ ab −q 2 1 1 +Π(q 2 ) , . Here the Landau gauge corresponds to ξ = 0, and the Feynman gauge to ξ = 1. We note that this convention differs from that in the widely used Form version [62] of the Mincer program [63] for three-loop self-energies, where the symbol xi represents 1 − ξ.

Triple-gluon vertex
Without loss of generality, one can set the momentum of the third gluon to zero, as depicted in figure 2. Then the triple-gluon vertex can be written in the following form: Γ abc µνρ (q, −q, 0) = −igf abc (2g µν q ρ − g µρ q ν − g ρν q µ ) T 1 (q 2 ) − g µν − q µ q ν q 2 q ρ T 2 (q 2 ) , (2.7) where g is the coupling constant and f abc are the structure constants of the gauge group. The first term in the square bracket corresponds to the tree-level vertex while the second term arises from radiative corrections, i.e., at the tree-level the form factors T 1,2 (q 2 ) read Because of Furry's theorem [64] and the fact that we have no colour-neutral particles, symmetric invariants with an odd number of indices cannot occur for internal fermion lines. Neither can such invariants occur for the adjoint representation. Hence, if we project out a d abc structure, we would get a scalar invariant with an odd number of f tensors, and such a combination must be zero. This has been checked explicitly to the equivalent of six-loop vertices in ref [65]. Due to the bosonic property of gluons, the totally antisymmetric colour factor f abc leads to antisymmetric Lorentz structure as in eq. (2.7). One could consider another Lorentz structure, − igf abc q µ q ν q ρ T 3 (q 2 ). (2.9) However, a Slavnov-Taylor identity requires T 3 (q 2 ) to vanish.

Ghost-gluon vertex
Since the tree-level vertex is proportional to the outgoing ghost momentum, nullifying this momentum gives identically zero in perturbation theory. Therefore, we only have two possibilities to set one of the external momenta to zero. One is the incoming ghost momentum and the other is the gluon momentum (figure 3): The subscript 'h' ofΓ h (q 2 ) indicates the function with vanishing incoming ghost momentum, whereas 'g' ofΓ g (q 2 ) denotes the vanishing gluon momentum. These functions are equal to one at the tree-level,Γ h (q 2 ) tree =Γ g (q 2 ) tree = 1. (2.12)

Quark-gluon vertex
We consider the case of a vanishing incoming quark momentum and the case of a vanishing gluon momentum (figure 4). Nullifying the outgoing quark momentum gives the same result as nullifying the incoming quark momentum. Then the vertex can be written as (2.14) T ij is the generator of the representation for the quark. The subscript 'q' indicates the functions with vanishing incoming quark momentum and 'g' indicates those with vanishing gluon momentum. At the tree-level we have

Renormalization
In a generic renormalization scheme R, the respective renormalizations of the gluon, ghost and quark fields can be written as The superscript 'B' indicates a bare quantity and 'R' a renormalized one. For the coupling constant, we define a = α s /(4π) = g 2 /(16π 2 ). Then a and the gauge parameter ξ are renormalized in dimensional regularization (D = 4 − 2ε) as follows: Here µ is the 't Hooft mass scale. We have used the fact that the gauge parameter is also renormalized by the gluon field renormalization constant, Z R ξ = Z R 3 . The renormalization of the self-energies and vertex functions is performed as and where the vertex renormalization constants are related to the field and coupling renormalization constants via the Slavnov-Taylor identities by (2.28) In MS-like schemes, the renormalization constants contain only pole terms with respect to ε and thus take the form The coefficients Z MS,(l,n) i are determined order by order in such a way that any renormalized Green's function becomes finite. The choice of the subtraction point defines the specific (MS-like) renormalization scheme; the beta function is independent of this choice.

Computations and checks
The results in this article are obtained by direct computation using the Forcer package [37][38][39], written in Form [66][67][68]. Forcer is a four-loop equivalent of Mincer [62,63]: for each topology a parametric reduction to simpler topologies is provided. Most topologies can be reduced using direct integration of one-loop insertions, by the triangle rule [69,70] or by the diamond rule [71]. If these substructures are absent, a manually designed reduction scheme is required. Through parametric integration-by-parts (IBP) identities [69,70], similar to the triangle rule, such reductions are found in a computer-assisted manner.
Great care has to be taken to prevent term blow-up from rewriting momenta to a new basis for the transitions between topologies. In Mincer this was done manually, but in Forcer a basis is found that minimizes the terms that are created automatically. The performance of the Forcer program has been demonstrated by computing the four-loop beta function in ten minutes in the Feynman gauge and in eight hours with all powers of the gauge parameter on a modern 24-core machine [37,38].
The diagrams have been generated using Qgraf [72]. Subsequently, we filter propagator insertions. These insertions are known lower-loop propagators, and can be factorized out of the diagram, see ref. [73]. Next, the topology is mapped to a built-in Forcer topology, after nullifying a leg for the vertices, and the colour factor is determined using the Form program of ref. [65]. To extract the form factors defined above, a generalization of the projection operators in ref. [40] to a generic gauge group is used. Then the Feynman rules are applied. The remaining Lorentz-scalar integrals (which include loop-momenta in numerators) are computed by the Forcer program.
The computation time varied between an hour and a week, on a single computer. The easy cases, such as the ghost propagator and quark propagator took an hour. The gluon propagators and ghost-ghost-gluon vertex and quark-quark-gluon vertex took about eight hours per configuration. The triple gluon vertex was the hardest case and took a week per configuration on a single machine with 24 cores. Had we chosen to compute with an expansion in ε, the computations would have been much faster.
We have checked our setup and results in various ways: • The longitudinal component of the gluon self-energy δ ab q µ q ν Π L (q 2 ) was shown to be zero by an explicit calculation at the four-loop level.
• The form factor T 3 (q 2 ) of the triple-gluon vertex in eq. (2.9) was computed and indeed vanished at the four-loop level.
• All the self-energies and vertex functions computed in this work were compared up to three loops with those in ref. [40]. Note that the finite parts of the vertex-function results in ref. [40] are only correct for SU(N ) gauge groups, since the presence of quartic Casimir operators was not taken into account in the reconstruction of the general case. This fact was also noted in ref. [20].
• The four-loop renormalization constants and anomalous dimensions for the case of SU(N ) and a general linear covariant gauge were provided in ancillary files [74] of ref. [13]. Directly after Forcer was completed, we established agreement with those results. For a generic group our results are in agreement with ancillary files of ref. [20].
• We remark that the ghost-gluon vertex is unrenormalizedZ MS = 1 in the Landau gauge. Moreover, our results confirm that the vertex has no radiative corrections when the incoming ghost momentum is nullified (i.e.,Γ MS h = 1) in the Landau gauge up to four loops. The renormalized results for all self-energies and vertex functions are rather lengthy and are thus given in appendix A. The unrenormalized results in terms of master integrals, and the values of the master integrals in 4−2ε dimensions are available online on arxiv.org, as ancillary files to the paper. A description of the files is given in appendix C.

Five-loop Landau-gauge QCD beta function in the MiniMOM scheme
In the MiniMOM scheme [56], the self-energies are completely absorbed into the field renormalization constants at the subtraction point q 2 = −µ 2 : Here the superscript 'MM' indicates a quantity in the MiniMOM scheme. In addition, motivated by the non-renormalization of the ghost-gluon vertex in the Landau gauge [54], the vertex renormalization constant for this vertex is chosen the same as that in MS, which is equal to one in the Landau gauge.
The above renormalization conditions lead to the following relations for the coupling constant and gauge parameter in the two schemes: . The scale dependence of the coupling constant in eq. (4.5) in this scheme is given by where we have used the beta function and gluon field anomalous dimension in MS, Note that the right-hand side of eq. (4.5), and hence that of eq. (4.7), is naturally given in terms of a MS and ξ MS . One has to convert them into a MM and ξ MM by inverting the series of eq. (4.5) and by using eq. (4.6). 2 Having results for the four-loop self-energies in the MS scheme at hand, one can obtain the five-loop beta function in the MiniMOM scheme from the five-loop beta function [16,19] and the four-loop gluon field anomalous dimension in the MS scheme. The result for SU (3) in the Landau gauge (ξ MM = 0) reads The result for a generic group and in an arbitrary covariant linear gauge can be found in appendix B; it agrees with the result given in ref. [57] up to four loops. As is well known, the first coefficient β MM 0 is scheme independent. The next coefficient β MM 1 has a gauge dependence and the universal value is obtained only in the Landau gauge. The last coefficient β MM 4 is the new result. In the MS scheme, some of higher values of the zeta function (e.g., ζ 2 3 , ζ 6 and ζ 7 at five loops) do not occur, for a discussion of this issue see refs. [21,35,75]. In contrast, one cannot expect their absence in the MiniMOM scheme. Indeed eq. (4.11) includes terms with ζ 2 3 and ζ 7 , and for ξ MM = 0 also ζ 6 occurs. The numerical values of the above beta function for three to five quark flavours arẽ  whereβ ≡ β(a)/(−β 0 a 2 ) has been re-expanded in powers of α s = 4πa. These values may be compared with those in the MS scheme [16,19] reading Obviously, the MiniMOM coefficients in eqs. (4.12) are (much) larger than their MS counterparts in eqs. (4.13) starting from the second order; moreover, they exhibit a definite growth with the order that is absent in the MS case. One may expect that this behaviour, and the larger value of α MM s , is more than compensated by smaller expansion coefficients for observables, leading to a better overall convergence in MOM-like schemes. However, this issue has been studied up to four loops in some detail for the R-ratio in electron-positron annihilation, without arriving at such a clear-cut conclusion [59].

Summary and outlook
We have computed the four-loop three-particle vertices with one vanishing momentum and the four-loop self-energies for QCD-like theories in a manner that is as general as presently possible. Our results, explicitly presented in the appendix of this article for the D = 4 renormalized quantities in the MS scheme, and available online in terms of their more lengthy bare counterparts, should be useful for precision studies of QCD-like theories with any simple compact gauge group, in any linear covariant gauge, for any MS-like or MOMlike renormalization scheme (see also ref. [40]), and in any number of space-time dimensions D. The latter requires replacing the master integrals, which is the only component that is not known exactly in D.
As an example application, we have determined the five-loop beta function in the MiniMOM scheme of ref. [56], i.e., we have extended the result of refs. [56,57] by one order in the coupling constant α s . This function appears to have a higher-order structure quite different from that in the MS scheme, thus inviting further studies especially for the physical case of QCD in four dimensions. Our computations have been made possible by the construction of Forcer [37][38][39], a four-loop generalization of the well-known Mincer program [62,63] for the parametric reduction of three-loop massless self-energy integrals. We have verified that, except for an issue in the transition from SU(N ) to a general gauge group that was also noted in ref. [20], our renormalized self-energies and vertices up to three loops agree with the results of ref. [40]. Furthermore, we have verified to four loops that the ghost-ghost-gluon vertex is unrenormalized in the Landau gauge, i.e., its anomalous dimension is proportional to ξ and that the gluon propagator is transverse. Earlier, we had checked the four-loop beta function and the four-loop SU(N ) [74] renormalization constants and anomalous dimensions of ref. [13].
Performing a similar calculation at five loops is a formidable challenge: so far most methods used for computations at five loops use infrared rearrangement, which modifies the finite terms. A direct computation, as has been performed in this article, would require a five-loop Forcer equivalent. This is hard for at least two reasons: the number of topologies that need a manually designed step-by-step IBP reduction is larger than 200, and the number of parameters increases from 14 at four loops to 20 at five loops. Additionally, given the size of the step from three to four loops, the required computer time could be an issue.

Acknowledgments
We would like to thank J.A. Gracey

A Four-loop MS results for self-energies and vertices
We expand the results for the scalar 'form factors' of the self-energies and vertices addressed in eqs. (2.1) -(2.16) at the point q 2 = −µ 2 in the MS scheme as Here the coupling constant a = α s /(4π) = g 2 /(16π 2 ) and the gauge parameter ξ are the renormalized ones in the MS scheme, i.e., a MS (µ 2 ) and ξ MS (µ 2 ). The Landau gauge corresponds to ξ = 0. Recall that this differs from the convention in Mincer and Forcer, where ξ = 0 corresponds to the Feynman gauge and ξ = 1 to the Landau gauge.
A.1 Gluon self-energy                                        where a and ξ are understood as a MS (µ 2 ) and ξ MS (µ 2 ). The coefficients c (l) are given by     Table 3. Symbols for renormalized results.