Heavy quark and gluino potentials to two loops

The static potentials for systems of a heavy quark and anti-quark, two gluinos and a quark and a gluino are computed for all possible colour configurations of a $SU(N_c)$ gauge group.

The potential energy between two heavy quarks is one of the fundamental quantities of the strong interaction and has been in the focus of the theoretical investigations already in the early days of QCD [1].The potential arises in a natural way when considering the nonrelativistic limit of a heavy quark and anti-quark system as an ingredient of the resulting Schrödinger-like equations (see Ref. [2] for a review).Thus, the potential constitutes a crucial input whenever the production of heavy particles is considered at threshold or bound state properties are calculated.Examples of Standard Model processes are the production of top quark pairs in electron positron collisions for a center-of-mass energy in the vicinity of twice the mass or the invariant-mass distribution of t t pairs at hadron colliders.Furthermore, one should also mention the evaluation of the energy levels and corrections to the wave function for heavy quark bounds states like the Υ or Ψ systems.
As far as processes beyond the Standard Model are concerned there have been recent publications where bound states of two gluinos, the massive super partners of the gluons, have been examined.Again, the corresponding potential, which has been used to two-loop order, plays a crucial role [3,4].Similarly, in Ref. [5] the threshold production of a gluinosquark pair is considered.The required potential can be obtained from the quark-gluino potential which is discussed below.
In this Letter we systematically compute the potentials of all colour configurations of a quark-anti-quark, gluino-gluino and quark-gluino bound state.To be precise, we consider the heavy-particle systems given in Tab. 1 and compute the potentials for the corresponding colour decomposition.bound state colour representation irreducible representations Table 1: Heavy-particle systems and their colour decomposition into irreducible representations.The subscripts "S" and "A" distinguish the symmetric and anti-symmetric octet representations.
Note that in our framework both the heavy quark q and gluino g are treated as external static colour sources added to the (massless) dynamical degrees of freedom of QCD.Thus except for colour there is no difference in the treatment of the gluino and the quark.As a consequence the potential of an anti-quark and a gluino is identical to the qg potential.
One comment concerning the colour decomposition of the gg potential is in order: As it is common practice we consider only the combination of 10 ⊕ 10.Furthermore, the colour structure R 7 is only non-vanishing for N c = 3 and thus it is not relevant for QCD [6][7][8][9][10].
We define the various potentials introduced above as follows where ij ∈ {q q, gg, qg} and c defines the colour state as given in Tab. 1.The renormalization scale is set to µ 2 = q 2 to suppress the trivial renormalization group terms on the r.h.s. of Eq. ( 1).For the potential V [10] gg we have to modify Eq. ( 1) slightly since there is no tree and one-loop contribution.Thus we write In Eq. ( 1) the coefficients a 1 and a 2 are the one-and two-loop corrections which are already present in the singlet contribution of the q q potential.They have been computed in Refs.[11][12][13][14][15] and can be found in Ref. [16] including higher order terms in (d − 4) (where d is the space-time dimension).The three-loop coefficient a 3 has been computed in Refs.[16][17][18].In less than four dimensions the static potential has recently been studied in Ref. [19], see also [20], and the N = 4 supersymmetric Yang-Mill theory has been considered in Ref. [21].
At tree-level and at one-loop order the only difference among the various potentials is due to the overall colour factor.At two-loop order we have introduced the quantity δa 2,ij which parametrizes the difference to the singlet result.It is currently only known for V [8] q q [22].Furthermore, also for V [1] gg the two-loop corrections have been computed [4,23] with the result δa [c] 2,gg = 0. (In Ref. [23] also the three-loop term of V [1] gg has been evaluated.)In this Letter we present the two-loop results for all remaining potentials listed in Tab. 1.
For the calculation we have employed standard techniques which include the automatic generation of the diagrams (see Fig. 1), the classification into different families of integrals, the application of projectors [6][7][8][9][10] and the reduction to master integrals using the Laporta algorithm [24][25][26].The latter have been taken over from Ref. [27].The colour factors have been computed with the help of the program color [28].We have performed the calculation for general gauge parameter and have checked that it drops out in the final result.
The standard techniques for the evaluation of the loop integrals appearing at one and two loops (see, e.g., Refs.[20,27]) can only be applied in a straightforward way to the singlet case since there, apart from light-fermion contributions, only the maximally non-Abelian parts contribute.In particular, no diagrams involving pinches occur, i.e. the integrals do not contain propagators of the form 1/(k 0 + i0) × 1/(k 0 − i0) where k is a loop momentum.Sample diagrams are shown in Figs.1(c) and (d) at one-loop and Figs.1(g) and (h) at two-loop order.However, for the non-trivial colour configurations also contributions involving pinches (see, e.g., Figs.1(b), (e) and (f)) have to be taken into account.The results of the corresponding diagrams are contained in the quantity δa 2,ij .We evaluate the integrals by either exploiting the exponentiation of the colour singlet potential or by carefully evaluating the potential in coordinate space starting from the Wilson loop definition.Both methods are described in detail in Refs.[20,22].In this Letter we have checked that they lead to the same result.
Table 2: Results for the C [c] and δa In Tab. 2 we present our results for C [c] and δa 2,ij for SU(N c ).Note that δa 2 is zero for the singlet contributions but also for gluino bound states in the symmetric octet configuration.One furthermore obtains a vanishing result for the representation 6 (qg) when specifying to QCD, i.e., setting N c = 3.It is remarkable that all non-vanishing contributions are proportional to the same combination (π 4 − 12π 2 ) with a prefactor depending on the colour state although the individual diagrams contributing to δa As far as the numerical importance of δa 2 is concerned one can compare the results in the last column of Tab. 2 with a 2 for the bottom and top system given by a 2 (n l = 4) ≈ 211.1 , a 2 (n l = 5) ≈ 155.8 .
(3) (The corresponding numbers for a 1 are 5.889 and 4.778, respectively.)In some cases one observes a significant reduction of the two-loop coefficient (see, e.g., the 27 configuration for n l = 4 where a 2 + δa [10] 2,gg ≈ 0.8) whereas in other cases the large value of a 2 is even further increased.
To conclude, in this Letter the quark-anti-quark, gluino-gluino and quark-gluino potentials have been computed for all possible colour configurations up to two loops.In all cases it is possible to identify the two-loop coefficient a 2 originating from the quark-anti-quark singlet potential.The additional contributions are given by a colour factor times (π 4 − 12π 2 ).

Figure 1 :
Figure 1: Sample diagrams contributing to V [c] ij at tree-level (a), one-loop (b)-(d) and two-loop order (e)-(h).The straight lines correspond to quarks or gluinos, respectively, and the curly lines represent gluons.
for the various colour configurations.In the right column we set N c = 3 and evaluate δa [c] 2,ij numerically.Note that c = 10 refers to the combination 10 ⊕ 10.Furthermore, R 7 has dimension zero for N c = 3.
do not show this proportionality and furthermore also contain terms without π 2 or π 4 .