Heavy quark and gluino potentials to two loops
aa r X i v : . [ h e p - ph ] S e p SFB/CPP-11-36TTP11-18
Heavy quark and gluino potentials to two loops
Tim Collet and Matthias Steinhauser
Institut f¨ur Theoretische TeilchenphysikKarlsruhe Institute of Technology (KIT)76128 Karlsruhe, Germany
Abstract
The static potentials for systems of a heavy quark and anti-quark, two gluinosand a quark and a gluino are computed for all possible colour configurations of a SU ( N c ) gauge group.PACS numbers: 12.38.Bx, 12.38.-t, 14.65.-q The potential energy between two heavy quarks is one of the fundamental quantities ofthe strong interaction and has been in the focus of the theoretical investigations already inthe early days of QCD [1]. The potential arises in a natural way when considering the non-relativistic limit of a heavy quark and anti-quark system as an ingredient of the resultingSchr¨odinger-like equations (see Ref. [2] for a review). Thus, the potential constitutes acrucial input whenever the production of heavy particles is considered at threshold orbound state properties are calculated. Examples of Standard Model processes are theproduction of top quark pairs in electron positron collisions for a center-of-mass energyin the vicinity of twice the mass or the invariant-mass distribution of t ¯ t pairs at hadroncolliders. Furthermore, one should also mention the evaluation of the energy levels andcorrections 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 recentpublications 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-looporder, plays a crucial role [3, 4]. Similarly, in Ref. [5] the threshold production of a gluino-squark pair is considered. The required potential can be obtained from the quark-gluinopotential which is discussed below.In this Letter we systematically compute the potentials of all colour configurations of aquark-anti-quark, gluino-gluino and quark-gluino bound state. To be precise, we considerthe heavy-particle systems given in Tab. 1 and compute the potentials for the correspond-ing colour decomposition.ound state colour representation irreducible representations q ¯ q ⊗ ¯3 1 ⊕ g ˜ g ⊗ ⊕ S ⊕ A ⊕ ⊕ ⊕ ⊕ R q ˜ g ⊗ ⊕ ⊕ q and gluino ˜ g are treated as externalstatic colour sources added to the (massless) dynamical degrees of freedom of QCD. Thusexcept for colour there is no difference in the treatment of the gluino and the quark. Asa consequence the potential of an anti-quark and a gluino is identical to the q ˜ g potential.One comment concerning the colour decomposition of the ˜ g ˜ g potential is in order: As itis common practice we consider only the combination of 10 ⊕
10. Furthermore, the colourstructure R is only non-vanishing for N c = 3 and thus it is not relevant for QCD [6–10].We define the various potentials introduced above as follows V [ c ] ij ( µ = ~q ) = − C [ c ] πα s ( ~q ) ~q " α s ( ~q )4 π a + (cid:18) α s ( ~q )4 π (cid:19) (cid:16) a + δa [ c ]2 ,ij (cid:17) , (1)where ij ∈ { q ¯ q, ˜ g ˜ g, q ˜ g } and c defines the colour state as given in Tab. 1. The renormal-ization scale is set to µ = ~q to suppress the trivial renormalization group terms on ther.h.s. of Eq. (1). For the potential V [10]˜ g ˜ g we have to modify Eq. (1) slightly since there isno tree and one-loop contribution. Thus we write V [10]˜ g ˜ g ( µ = ~q ) = − πα s ( ~q ) ~q (cid:18) α s ( ~q )4 π (cid:19) δa [10]2 , ˜ g ˜ g . (2)In Eq. (1) the coefficients a and a are the one- and two-loop corrections which arealready present in the singlet contribution of the q ¯ q potential. They have been computedin Refs. [11–15] and can be found in Ref. [16] including higher order terms in ( d − d is the space-time dimension). The three-loop coefficient a has been computedin Refs. [16–18]. In less than four dimensions the static potential has recently beenstudied in Ref. [19], see also [20], and the N = 4 supersymmetric Yang-Mill theory hasbeen considered in Ref. [21].At tree-level and at one-loop order the only difference among the various potentials is dueto the overall colour factor. At two-loop order we have introduced the quantity δa [ c ]2 ,ij whichparametrizes the difference to the singlet result. It is currently only known for V [8] q ¯ q [22].2 a) (b) (c) (d)(e) (f) (g) (h) Figure 1: Sample diagrams contributing to V [ c ] ij at tree-level (a), one-loop (b)–(d) andtwo-loop order (e)–(h). The straight lines correspond to quarks or gluinos, respectively,and the curly lines represent gluons.Furthermore, also for V [1]˜ g ˜ g the two-loop corrections have been computed [4, 23] with theresult δa [ c ]2 , ˜ g ˜ g = 0. (In Ref. [23] also the three-loop term of V [1]˜ g ˜ g has been evaluated.) Inthis 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 automaticgeneration of the diagrams (see Fig. 1), the classification into different families of integrals,the application of projectors [6–10] and the reduction to master integrals using the Laportaalgorithm [24–26]. The latter have been taken over from Ref. [27]. The colour factorshave been computed with the help of the program color [28]. We have performed thecalculation for general gauge parameter and have checked that it drops out in the finalresult.The standard techniques for the evaluation of the loop integrals appearing at one andtwo loops (see, e.g., Refs. [20, 27]) can only be applied in a straightforward way to thesinglet case since there, apart from light-fermion contributions, only the maximally non-Abelian parts contribute. In particular, no diagrams involving pinches occur, i.e. theintegrals do not contain propagators of the form 1 / ( k + i × / ( k − i
0) where k isa loop momentum. Sample diagrams are shown in Figs. 1(c) and (d) at one-loop andFigs. 1(g) and (h) at two-loop order. However, for the non-trivial colour configurationsalso contributions involving pinches (see, e.g., Figs. 1(b), (e) and (f)) have to be takeninto account. The results of the corresponding diagrams are contained in the quantity δa [ c ]2 ,ij . We evaluate the integrals by either exploiting the exponentiation of the coloursinglet potential or by carefully evaluating the potential in coordinate space starting fromthe Wilson loop definition. Both methods are described in detail in Refs. [20, 22]. In thisLetter we have checked that they lead to the same result.3 j c C [ c ] δa [ c ]2 ,ij δa [ c ]2 ,ij ( N c = 3) q ¯ q ( N c − N c − N c N c ( π − π ) − . g ˜ g N c S N c A N c − π − π ) 126 .
210 — − N c ( π − π ) 94 . − ( N c + 2) ( N c + 1) ( π − π ) − . R ( N c −
2) ( N c −
1) ( π − π ) — q ˜ g N c − ( π − π ) 21 .
12 12 N c ( N c −
3) ( π − π ) 015 −
12 12 N c ( N c + 3) ( π − π ) − . C [ c ] and δa [ c ]2 ,ij for the various colour configurations. In the rightcolumn we set N c = 3 and evaluate δa [ c ]2 ,ij numerically. Note that c = 10 refers to thecombination 10 ⊕
10. Furthermore, R has dimension zero for N c = 3.In Tab. 2 we present our results for C [ c ] and δa [ c ]2 ,ij for SU ( N c ). Note that δa is zero for thesinglet contributions but also for gluino bound states in the symmetric octet configuration.One furthermore obtains a vanishing result for the representation 6 ( q ˜ g ) when specifyingto QCD, i.e., setting N c = 3. It is remarkable that all non-vanishing contributions areproportional to the same combination ( π − π ) with a prefactor depending on thecolour state although the individual diagrams contributing to δa [ c ]2 ,ij do not show thisproportionality and furthermore also contain terms without π or π .As far as the numerical importance of δa is concerned one can compare the results inthe last column of Tab. 2 with a for the bottom and top system given by a ( n l = 4) ≈ . ,a ( n l = 5) ≈ . . (3)(The corresponding numbers for a are 5 .
889 and 4 . n l = 4 where a + δa [10]2 , ˜ g ˜ g ≈ .
8) whereas in other cases the large value of a is evenfurther increased.To conclude, in this Letter the quark-anti-quark, gluino-gluino and quark-gluino potentials4ave been computed for all possible colour configurations up to two loops. In all casesit is possible to identify the two-loop coefficient a originating from the quark-anti-quarksinglet potential. The additional contributions are given by a colour factor times ( π − π ). Acknowledgements
We would like to thank Matthias Kauth, Johann K¨uhn and Alexander Penin for manyuseful discussions and communications. This work was supported by the DeutscheForschungsgemeinschaft through the SFB/TR-9 “Computational Particle Physics”.
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