Fermionic contributions to the three-loop static potential

We consider the three-loop corrections to the static potential which are induced by a closed fermion loop. For the reduction of the occurring integrals a combination of the Gr\"obner and Laporta algorithm has been used and the evaluation of the master integrals has been performed with the help of the Mellin-Barnes technique. The fermionic three-loop corrections amount to 2% of the tree-level result for top quarks, 8% for bottom quarks and 27% for the charm quark system.


Introduction
The potential between a heavy quark and its anti-quark is a crucial quantity both for understanding fundamental properties of QCD, such as confinement, and for describing the rich phenomenology of heavy quarkonia [1] (see also Ref. [2] for a recent review about the static potential).
Within perturbation theory the static potential can be computed as an expansion in the strong coupling α s and the inverse heavy-quark mass or, equivalently, in the heavy-quark velocity v. The leading order result in v is known up to the two-loop approximation [3][4][5][6][7][8] which has been completed about ten years ago. For the three-loop corrections there are only estimates relying on Padé approximations [9] or based on renormalon studies [10].
A new feature of the three-loop corrections is the appearance of an infrared divergence which was discussed for the first time in Ref. [11]. A quantitative analysis of this effect can be found in Ref. [12] (see also Ref. [13]) where a proper definition of the static potential within perturbation theory is provided. Furthermore, it is argued that the infrared singularities cancel in physical quantities after including the contribution where so-called ultra-soft gluons interact with the heavy quark anti-quark bound state (see also, e.g., Refs. [14][15][16]). Higher order logarithmic contributions to the infrared behaviour of the static potential have been considered in Refs. [17,18].
In this paper we compute the fermionic contribution to the three-loop static potential which is infrared safe. Partial results have already been published in Refs. [19,20].
The static potential enters as a building block in a variety of physical quantities. Often at three-loop order only estimations are used or the three-loop coefficient -usually called a 3 -appears as a parameter in the final result. Let us in this context mention the determination of the bottom and top quark mass from the ground state energy of the heavy quark system which has been computed to third order in Ref. [16]. The error on the mass values due to the unknown three-loop coefficient amounts to 14% (13%) of the total uncertainty for the bottom (top) quark. Similarly, a 3 enters the calculation of the total cross section for top quark threshold production at next-to-next-to-next-to leading order (see, e.g., Ref. [21]). The static energy between two heavy quarks has often been used in order to compare perturbative calculations with simulations on the lattice (see, e.g., Refs. [22][23][24][25]). Also in this context the knowledge of a 3 is crucial and is expected to lead to a better agreement between the two approaches [24]. Last not least let us mention the extraction of the strong coupling from lattice simulations where again the static potential plays a crucial role [26,27] and the knowledge of a 3 would be highly desirable.
Let us for completeness mention that the one-loop mass-suppressed corrections to the static potential have been evaluated in Refs. [28][29][30][31][32][33], the two-loop terms in Ref. [8]. Light quark mass effects have been considered in Ref. [34]. A collection of all relevant contributions needed up to next-to-next-to-next-to-leading order can be found in Ref. [15]. Furthermore, the two-loop corrections for the case where the quark and anti-quark form Figure 1: Sample diagrams contributing to the static potential at tree-level, one-, two-and three-loop order. In this paper only the fermionic corrections are considered at three-loop order which excludes diagrams of type (h).
an octet state have been evaluated in Ref. [35].
The remainder of the paper is organized as follows: In the next Section we present details of our calculation. In particular we discuss the various types of Feynman diagrams which occur at three-loop order and their contributions to the individual colour factors. In Section 3 our results are presented and Section 4 contains our conclusions.

Calculation
In the practical calculation of quantum corrections to the static potential one has to consider a heavy quark and its anti-quark which interact via the exchange of gluons. In Fig. 1 some sample diagrams up to three-loop order are shown.
In momentum space the static potential can be cast into the form where α s denotes the strong coupling in the MS scheme and explicit results for a 1 [3,4] and a 2 [5][6][7][8] are given below in Eq. (4). The infrared logarithm at order α 3 s follows the conventions of Ref. [15] and the renormalization group logarithms ln(µ 2 / q 2 ) can be recovered with the help of where L = ln(µ 2 / q 2 ) and the coefficients of the β function (see, e.g., Ref. [36]) read Here, are the eigenvalues of the quadratic Casimir operators of the adjoint and fundamental representations of the SU(N c ) colour gauge group, respectively, T F = 1/2 is the index of the fundamental representation, and n l is the number of light-quark flavours. Let us at this point only mention that a 1 has the colour structures C A and T F n l where the latter contribution originates from one-loop fermionic corrections to the gluon propagator. Note that there is no colour factor C F since this contribution is generated via an iteration of the tree-level result. Consider, e.g., the one-loop planar-ladder and the crossed-ladder diagram of Fig. 1(b) with the colour factors C 2 F and C 2 F − C A C F /2, respectively. It is easy to see (see, e.g., Ref. [37]) that the C 2 F term can be generated by iterations of the leading order diagram in Fig. 1(a) leaving only the C A C F term as genuine one-loop contribution.
Similarly, at two-loop order there are the colour factors C 2 A , C A T F n l , C F T F n l , and (T F n l ) 2 where the latter two originate from loop corrections to the gluon propagator connecting the quark and the anti-quark. a 2 contains no C 2 F and C F C A terms since their contribution is again generated by iterations of lower-order results.
The rule that a colour factor C F can only arise from corrections to a fermion bubble in a gluon line also holds at three-loop level. This requires a careful analysis of the colour factors for each class of diagram. For example, the colour factor of the graph in Fig. 1(e) receives a contribution 1 (C F −C A /2)T F n l from two-loop fermionic subdiagram and a factor (C F − C A /2) from the remaining crossed box structure. Whereas the complete first factor has to be taken into account only the C A term of the second factor contributes to the potential. In a similar manner all diagrams have to be analyzed which leads to the colour  3 . In this paper we compute all coefficients except the one of C 3 A . The results for the structures C A (T F n l ) 2 , C F (T F n l ) 2 and (T F n l ) 3 can be found in Refs. [19,20].
At three-loop order there is a new class of diagrams containing a "light-by-light" subdiagram (see Fig. 1(f) for a sample graph) which develops the colour factors d abcd F d abcd F /N A and C 2 A T F n l . Note that these contributions are not connected to iterations and are thus already present in QED (i.e. for C A = 0, d abcd F = 1, N A = 1 and T F = 1).
Since we use non-relativistic QCD as a starting point for the evaluation of the Feynman diagrams the momentum transfer between the quark and the anti-quark represents the only relevant scale in the problem. Thus all integrals can be mapped to the two-point functions which are shown in Fig. 2 in diagrammatical form. Next to purely massless lines originating from the gluon, ghost and light-quark propagators also static lines from the heavy quarks are present. The one-and two-loop diagrams have been extensively studied in Ref. [37][38][39]. As far as the three-loop diagrams are concerned one can perform a partial fractioning in those cases where three static lines meet at a vertex. This leads to many different three-loop graphs involving, however, at most three static lines. Thus any resulting integral is labeled by twelve indices one of which corresponds to an irreducible numerator.
Altogether we have to consider about 70 000 integrals (allowing for a general QCD gauge parameter ξ) which can all be mapped to one of the diagrams shown in Fig. 3. Thereby the linear propagators can appear in two variants: either in the form (−v · k − i0) or (−v · k + i0). If the loop momenta, k, l and r, in the upper row of Fig. 3 are chosen as the momenta of the three upper lines, then the first diagram appears in two ways: either with the product Figure 3: Three-loop diagrams of "ladder" (first three diagrams), "non-planar" (forth diagram in upper row) and "mercedes" type (lower row) which have to be considered for the fermionic part of a 3 .
or with the product For the calculation of the diagrams we proceed in the following way: They are generated with QGRAF [40] and further processed with q2e and exp [41,42] where a mapping to the diagrams of Fig. 2 is achieved. In a next step the reduction of the integrals is performed with the program package FIRE [43] which implements a combination of the Laporta [44] and the Gröbner algorithm (see, e.g., Ref. [45]). This leads us to about 100 master integrals which have to be evaluated in an expansion in ǫ with the help of the Mellin-Barnes technique. Non trivial examples are discussed in Refs. [19,20] where also explicit results are given. We managed to compute all but four coefficients of the ǫ expansion analytically. As a crucial tool providing very important numerical cross checks of the analytical results we applied the program FIESTA [46] which is a convenient and efficient implementation of the sector decomposition algorithm. Finally, let us mention that we evaluate the colour factors with the help of the program color [47].
In our calculation we allowed for a general gauge parameter ξ in the gluon propagator and checked that ξ drops out in the final result. This constitutes a strong check on the correctness of our result.
In order to obtain a finite result one has to renormalize the strong coupling which we perform within the MS scheme. The corresponding renormalization constant can, e.g., be found in Ref. [36].

Results
Let us in a first step present the results for the one-and two-loop coefficients including higher orders in ǫ since these terms are needed for the renormalization procedure. We obtain where the one-and two-loop results (in the limit ǫ → 0) can be found in Refs. [3,4] and [5][6][7][8], respectively. In Eq. (4) ζ is Riemann's zeta function, with the value ζ(3) = 1.202057 . . ..

The three-loop result can be cast in the form
where the first three coefficients on the right-hand side read where the SU(N c ) colour factors are given by  Table 1: Radiative corrections to the potential V (| q |) where the tree-level result is normalized to 1 (cf. Eq. (8)).
In Eq. (6) only the coefficient of d abcd F d abcd F is affected by the limited numerical precision of the four coefficients only known numerically which is indicated by the number in round brackets.
We are now in the position to briefly discuss the numerical effect of the new corrections. Inserting the results for a 1 , a 2 and a 3 in Eq. (1) it takes the form where the ellipses denote higher order terms and µ 2 = q 2 has been chosen in order to suppress the infrared logarithm. From Eq. (8) one observes that both at one-and twoloop order the linear n l term is negative and leads to a screening of the (positive) non-n l contribution by an amount of about 50% for n l = 5. Also at three-loop order the linear n l term is negative and has a sizeable coefficient. Both for a 2 and a 3 the n 2 l contribution is small; the same is true for the n 3 l term of a 3 . In Tab. 1 we present the one-, two-and three-loop results from the square bracket of Eq. (8) and choose n l according to the charm, bottom and top quark case. In the second column we also provide the numerical value of α s corresponding to the soft scale where µ ≈ m q α s (and m q is the heavy quark mass). It is interesting to note that the threeloop corrections computed in this paper lead to corrections which are of the same order of magnitude as the two-loop corrections, however, with a different sign. In fact, one obtains corrections of about −27%, −8% and −2% for charm, bottom and top quarks, respectively. Furthermore, let us mention that the unknown constant has to be of the order 10 4 (and positive) in order to significantly reduce the size of the three-loop corrections.
Let us now compare our explicit calculation with the predictions based on Padé approximation.
Finally we want to specify our result for V (| q |) to QED which describes the potential of two heavy leptons in the presence of n l massless leptons. Substituting for the colour factors C A = 0, C F = 1, T F = 1, d abcd F = 1 and N A = 1 we obtain whereᾱ =ᾱ( q 2 ) is the QED coupling in the MS scheme and after the second equality sign n l = 1 has been chosen. This corresponds to a bound state of a muon and an anti-muon in the presence of a massless electron pair. The coefficients in Eq. (9) are significantly smaller as in the case of QCD which results in corrections of the order 10 −8 from the three-loop term.

Conclusion and outlook
In this letter we report about the calculation of the fermionic corrections to the static potential of a quark and an anti-quark. All occurring integrals are reduced to about 100 master integrals with the help of the program FIRE. The main result can be found in Eqs. (6) where the three-loop coefficients are given for each occurring colour structure. The numerical corrections of the new three-loop terms are quite sizeable when applied to the system of two charm, bottom or top quarks. However, for a definite conclusion one has to wait for the n l independent three-loop coefficient a 3 is currently in progress. We do not expect any conceptual problems. However, significantly more Feynman diagrams contribute which leads to many new graphs in addition to those shown in Fig. 3. As a consequence also more master integrals have to be evaluated.