The logarithmic contribution to the QCD static energy at NNNNLO

Using pNRQCD and known results for the field strength correlator, we calculate the ultrasoft contribution to the QCD static energy of a quark-antiquark pair at short distances at NNNNLO in alpha_s. At the same order, this provides the logarithmic terms of the singlet static potential in pNRQCD and the log alpha_s terms of the static energy.


Introduction
The ground state energy, E 0 (r), of a static quark and a static antiquark separated by a distance r is a key object for the understanding of the QCD dynamics. It is also a basic ingredient of the Schrödinger-like formulation of heavy quarkonium systems [1]. Its linear behavior at long distances is a signal for confinement [2], but also at short distances (r ≪ 1/Λ QCD ), where weak coupling calculations are reliable, it shows a non-trivial behavior. Indeed, when calculated in perturbation theory, infrared divergences are found starting at three loops [3,4]. These are due to the virtual emission of ultrasoft gluons with energy of the order E 0 (r), which turn a color singlet quark-antiquark pair into a color octet one and vice-versa. The proper treatment of the ultrasoft emissions requires the resummation of an infinite class of diagrams, which produces a non-analytic dependence on α s (typically logarithms of it). We shall focus here on this short distance behavior.
The current knowledge of E 0 (r) at short distance may be summarized as follows where C F = T F (N 2 c − 1)/N c , C A = N c , T F = 1/2, N c is the number of colors, T F n f are the first two coefficients of the beta function, n f is the number of (massless) flavors, γ E is the Euler constant and α s is the strong coupling constant in the MS scheme. The one-loop coefficient a 1 is given by [5,6] and the two loop coefficient a 2 by [7,8,9,10] The logarithmic piece of the third-order correction was calculated in [11], whereas the non-logarithmic pieceã 3 has not been calculated yet. However,ã 3 is believed to be dominated by contributions which are known from renormalization group arguments [12]. If we writeã 3 = a 3 + a RG 3 , a RG 3 ≫ a 3 . a RG 3 has a known expression in terms of the coefficients of the beta function and of those entering in the potential at lower orders (see [12], whereã 3 = −48π 3 × V 3 , a 3 = 64c 0 ). Estimates of a 3 have been carried out using Padé approximations [12] and renormalon dominance [13,14,15,16], which are consistent with the inequality above, and give similar results. The double logarithmic fourth-order correction may be obtained from [17], where higher-order contributions of the form α n+3 s log n α s were resummed using renormalization group techniques. It reads The main result of this letter is the calculation of the logarithmic fourth-order correction to the singlet potential, a L 4 ; we anticipate that it reads The non-logarithmic pieceã 4 remains unknown.
A convenient method to calculate the logarithmic contributions to Eq. (1), which steam from the dynamics at the ultrasoft scale E 0 (r), consists in integrating out from static QCD degrees of freedom at the soft energy scale 1/r and working within the effective field theory framework of pNRQCD [18,19](see [20] for a review). The quark-antiquark system may be in a color singlet or in a color octet configuration, which are encoded in color singlet, S, and color octet, O, fields in pNRQCD. At leading order in the multipole expansion, the integration of the soft energy scale gives rise to a singlet, V s (r; µ), and an octet, V o (r; µ), static potential, which depend on r and a factorization scale µ. At next-to-leading order, two more "potentials" appear, V A (r; µ) and V B (r; µ), which are the matching coefficients of the singlet-octet and octet-octet vertices respectively. At this order, the pNRQCD Lagrangian reads where L light is the part of the Lagrangian involving gluons and light quarks, and coincides with the QCD one. The dots stand for higher-order terms in the multipole expansion. The static energy calculated from the above Lagrangian has the form where δ US (r, V s , V o , V A , V B , . . . ; µ) (δ US for short) contains contributions from the ultrasoft gluons. V s (r; µ) and V o (r; µ) do not depend on µ up to N 2 LO [19]. The former coincides with E 0 (r) at this order and the latter may be found in [21]. The fact that the µ dependence of δ US must cancel the one in V s (r; µ) is the key observation that leads to a drastic simplification in the calculation of the log α s terms in E 0 (r). So, for instance, the logarithmic contribution at N 3 LO, which is part of the three-loop contributions to V s (r; µ), may be extracted from a one-loop calculation of δ US [11,19] and the single logarithmic contribution at N 4 LO, which is part of the four-loop contributions to V s (r; µ), may be extracted from a two-loop calculation of δ US . 2 In Sec. 2, we review the calculation of the third-order logarithmic term since it follows the same lines as that of the fourth-order one, which will be presented in Sec. 3. In Sec. 4, we conclude and discuss some applications of this work.

Review of the third-order logarithmic correction
In d dimensions, the order r 2 contribution due to ultrasoft gluons reads [11,19] φ(y, x) adj ab is the Wilson line in the adjoint representation connecting the points y and x by a straight line (t stands for (t, 0)). We will evaluate Eq. (8) perturbatively in α s . The dependence on α s , apart from the trivial g 2 factor, enters through (i) the V s and V o potentials, (ii) V A and (iii) the field strength correlator of the chromoelectric fields.
Note that at leading and next-to-leading order V s and V o only differ by an overall color factor.
(ii) At tree level V A = 1. 2 We denote N n LO, contributions to the potential of order α n+1 s and N n LL, contributions of order α n+2 Leading-order contribution to the field strength correlator. The double line represents the gluonic string, the circled cross and the springy line the chromoelectric field correlator. Note that our convention differs from the one in [23], where the gluonic string is either not represented, if no gluons emerge from it, or is represented by a dashed line.
(iii) The two-point field strength correlator can be parameterized in terms of two scalar functions D(z 2 ) and D 1 (z 2 ) according to where z = y − x [22]. In (8), x and y only differ in the time component, hence z = t. Furthermore, in d dimensions, the chromoelectric component is given by The leading-order contribution to the field strength correlator is given by the diagram shown in Fig. 1.
Note that keeping ǫ = 0 in the chromoelectric correlator provides a regularization for the integral over t in Eq. (8).
We now insert (i), (ii) and (iii) into Eq. (8). Since at the ultrasoft scale t(V o −V s ) ∼ 1, the integral in t is performed without expanding the exponential and making use of The final result reads: where Note that the α s coming from the potential is evaluated at the soft scale 1/r, while the α s coming from the ultrasoft coupling is evaluated at the scale µ. This will become relevant in the next section. The ultraviolet divergence in (14) can be reabsorbed by a renormalization of the potential. In the MS scheme, in coordinate space, we have: Since the static energy is µ independent, from the calculation above we infer that the logarithmic contribution to V s (r; µ) at order α 4 s must be which added to the renormalized δ US contribution (from (14)) gives the log α s term displayed in the third line of Eq. (1). This term was first calculated in [11], where the cancellation between the IR cut-off of V s (r; µ) and the UV cut-off of the pNRQCD expression was checked explicitly by calculating the relevant Feynman diagrams in the Wilson loop.
A comment is in order concerning the scheme dependence of the calculation of δ US . This is not important if we are only interested in the logarithmic contribution, but it is if we wish eventually to combine our result with a (yet to be done) calculation of V s (r; µ) at N 3 LO and get the non-logarithmic pieces of the static energy right. We will assume that such a calculation will be done in momentum space and that dimensional regularization and the MS scheme will be used to renormalize the UV divergences, like in the N 2 LO calculation [7,8,9]. The result will still be IR divergent when d → 4, and the question is how one should proceed in order to combine that result with ours in a consistent way. 3 We propose to convert the (UV renormalized) momentum-space potential to coordinate space (in d dimensions) in that calculation, and together to use ddimensional expressions for all the objects in our calculation, namely also for V s (r; µ) and V o (r; µ) (V A (r; µ) remains the same in d dimensions). This guarantees that the IR behavior of the regulated effective theory is exactly the same as the one of the fundamental theory. Had we expanded V o − V s in Eq. (8) we would have obtained zero, which means that the UV divergences, which remain after renormalization by the MS QCD counterterms (and by that of the color octet field wave function) in the effective theory, cancel exactly the IR divergences. Therefore, as a consequence of the fact that the IR behavior of the regulated effective theory is the same as the one of the fundamental one, the UV divergences in (14) cancel exactly the IR divergences in V s (r; µ), the µ dependence disappears, and the non-logarithmic pieces are correctly calculated. This procedure would be analogous to the one employed in [24]. Alternatively, one could use MS for the IR divergences of V s (r; µ) in momentum space, work out the momentum space expressions for the d-dimensional version of (14) and make the MS UV subtraction accordingly.
In the following section, we will use the same procedure employed here to obtain the next-to-leading IR logarithmic dependence of the static potential. That is the logarithmic α 5 s contribution to the potential, which is part of the N 4 LO contribution.

Fourth-order logarithmic correction
Equation (8) does not rely on an expansion in α s , therefore it also provides NLO contributions to δ US . In fact, as we argue next, it provides the full contribution to this order.
In principle, we may have diagrams with more insertions of the operators in (6) and diagrams with operators of higher order in the multipole expansion that contribute to δ US at NLO. Concerning the former, for symmetry reasons we need at least two more operator insertions, which implies a suppression of α 3 s with respect to the leading-order δ US . Concerning the latter, operators of higher order in the multipole expansion may be found in [25,26]. Their contributions are suppressed by α 2 s with respect to the leading-order δ US . To see this just recall that the ultrasoft fields (and derivatives acting on them) must be counted as E 0 (r) ∼ α s /r. Then, any insertion of the kind dt r · E implies an α s suppression (with an extra α s suppression for any r i D j acting on the chromoelectric field). For a given diagram, additional suppressions may appear due to the coupling constants in front of the chromoelectric fields. The NLO contribution to δ US is then provided by Eq. (8) evaluated at relative order α s . Since the dependence in α s enters through V A , V s , V o and the chromoelectric correlator, we need the O(α s ) corrections to all these quantities. These will be given in the following two sections. Finally, in Sec. 3.3, we will obtain the fourth-order logarithmic correction to the potential.
The O(α s ) corrections to V s and V o are well known. In particular, we have The matching coefficient V A can be obtained by matching static QCD to pN-RQCD at order r in the multipole expansion. At leading order in α s , we have to calculate the diagrams shown in Fig. 2. They give the tree level result V A = 1. One may naively expect the first correction to be O(α s ), but it is not. 4 This becomes clear if we perform the calculation in dimensional regularization and in Coulomb gauge 5 . Indeed, the diagrams that we can draw at O(α s ) correspond either to self-energy corrections or to iterations of the Coulomb potential, which are identical in the effective theory and hence do not contribute to the matching. Then, the first non-vanishing correction to the tree level result may possibly come from diagrams like the one in Fig. 3, which is O(α 2 s ) and, therefore, unimportant here.

O(α s ) correction of the field strength correlator
The O(α s ) correction to the QCD field strength correlator was calculated in [23]. It is given by the diagrams in Fig. 4. Here we need the expression in d ¡ Fig. 3. Example of static QCD diagram that contributes to the next-to-next-to-leading order matching of V A .
dimensions because in (8) the integral over t is singular. The d-dimensional result for the α s correction is [27] with where B(u, v) = Γ(u)Γ(v)/Γ(u + v).
Since the external points x and y are fixed, the divergences that we encounter in D i0i0 coming from the expressions above should cancel against the vertex and gluon and octet field propagator counterterms. The counterterm for the vertex is zero, since, as seen in the previous section, the first correction to V A is of order α 2 s . The counterterm for the gluon propagator is the usual one in QCD. The counterterm for the octet propagator coincides with the counterterm for the quark propagator in the Heavy Quark Effective Theory [28] but with the quark in the adjoint representation. We can represent the counterterm contributions by the diagrams of Fig. 5. We have checked that: (i) the divergence coming from the first diagram in Fig. 4 is canceled by the counterterm of the gluon propagator, (ii) the diagram (b) of Fig. 4 does not give a divergent contribution (as one would expect from the fact that the gluons are attached to the external fixed points only) and (iii) when we sum the remaining diagrams the divergence that we obtain is exactly canceled by where in the brackets we have kept separated the −2 coming from the octet propagator counterterm from the − 5 3 + 4 3 T F n f N c coming from the gluon propagator one. The renormalized d-dimensional result for the α s correction to the chromoelectric correlator is which, indeed, is finite for ǫ → 0.

Calculation of the fourth-order logarithmic correction
The results of the two preceding sections provide all the necessary ingredients to compute δ US at NLO. Let us split δ US as follows where the first and second terms stand for the α s corrections to the field strength correlator and to the potentials respectively, and the last term accounts for the contribution induced by a change of scale in the N 3 LO calculation.
First, we shall consider the contribution (24) to the field strength correlator. After integration over t, which can be done using Eq. (13), we obtain with A = 1 24 γ E − 17 log 2 + 11 2 log 2 2 + 47 12 log π − 12ζ(3) Next, we display the contribution that we obtain if in (8) we use the leadingorder expression for the chromoelectric correlator but the O(α s ) correction for The ultraviolet divergences in the expressions (26) and (32) come from the integration over time in (8). They can be absorbed by a renormalization of the potential, analogous to (15).
Finally, we obtain another contribution if in the renormalized version of (14) we change α s (µ) to α s (1/r) (we want all α s evaluated at the scale 1/r): Adding up the renormalized versions of (26) and (32) and equation (33), we obtain the contribution of δ US to E 0 (r) at order α 5 s . The complete calculation of E 0 (r) at this order requires the knowledge of V s (r; µ) at the same order. However, to obtain the terms proportional to log α s it is enough to enforce E 0 (r) to be independent of the factorization scale µ. This constrains the terms α 5 s log 2 rµ and α 5 s log rµ of the singlet static potential to be Summing (34) with (25) provides the coefficients a L2 4 and a L 4 of the static energy E 0 (r) given in Eqs. (4) and (5) respectively.
Note that: (i) in order to cancel the µ dependence of the two double logarithms in δ US , log(rµ) log((V o −V s )/µ) and log 2 ((V o −V s )/µ), against the single double logarithm in δV s , log 2 rµ, the coefficient of log(rµ) log((V o − V s )/µ) must be twice the one of log 2 ((V o − V s )/µ). (ii) The coefficient of the double logarithm log 2 rµ in δV s should coincide with the one obtained expanding the renormalization group improved static potential of [17]. (iii) The coefficients a L2 4 and a L 4 must be renormalization scheme independent 6 . We have explicitly checked that our result satisfies these requirements.

Conclusions
We have calculated the ultrasoft contribution to the QCD static energy of a quark-antiquark pair at order α 5 s . This is sufficient to obtain the logarithmic contribution to the pNRQCD singlet static potential at N 4 LO, which, in turn, provides the α 5 s log 2 α s /r and α 5 s log α s /r terms of the static energy of a quarkantiquark pair at distance r. The calculation heavily relies on effective field theory techniques and uses the result of Ref. [23] as a key ingredient.
Possible applications of the result include precision comparisons with lattice data, heavy quarkonium spectra and t-t production near threshold.
At short distances, the perturbative expression of the QCD static energy has been compared with lattice data at N 2 LO in [30,31] and at N 2 LL in [14]. Our analysis provides a key ingredient for a N 3 LL analysis.
Starting from the N 3 LO in α s , the quarkonium mass becomes sensitive to the ultrasoft scale, if the ultrasoft scale is assumed to be much larger than Λ QCD [32,33,24,34]. Our result also provides an important ingredient for the calculation of the quarkonium mass at N 3 LL accuracy.
Top-quark pair production near threshold, which will become an important production process at the ILC, is presently known at N 2 LO [35]. The cross section at N 2 LL (see e.g. [36,37]) and at N 3 LO (see e.g. [38,39]) is computed presently by several different groups. Our result will contribute to the cross section at N 3 LL. The third-order renormalization group improved expression will be needed to resum logarithms potentially as large as the N 3 LO and reduce the scale dependence of the cross section.