The static quark potential to three loops in perturbation theory Alexander V. Smirnov a, Vladimir A. Smirnov b, Matthias Steinhauser c a Scientific Research Computing Center of Moscow State University, Russia b Nuclear Physics Institute of Moscow State University, Russia c Institut für Theoretische Teilchenphysik, Karlsruhe Institute of Technology, Germany The static potential constitutes a fundamental quantity of Quantum Chromodynamics. It has recently been evaluated to three-loop accuracy. In this contribution we provide details on the calculation and present results for the 14 master integrals which contain a massless one-loop insertion. 1. Introduction The static potential enters a variety of observables connected to heavy-quark physics. Among them are the prominant examples like the determination of the bottom quark mass from Υ sum rules or the cross section for the top quark pair production close to theshold. It is desirable for both quantities to perform a third-order analysis which requires the evaluation of the static quark potential to three loops. In order to fix the notation we write the static potential in momentum space in the following form [ V q 4πC Fα s q q 1 α s q 4π a 1 αs q αs q a a 4π 4π ] 8π CA ln µ q. 1 where C A N c and C F N c 1/N c. In Eq. 1 we identify the renormalization scale µ and the momentum transfer q. The complete dependence on µ can easily be restored with the help of Eq. of Ref. [1]. The one- and two-loop coefficients a 1 [,] and a [4 7] are given in Eq. 4 of Ref. [1] where Talk presented at Loops and Legs in Quantum Field Theory 010, Wörlitz, Germany, April 5-0, 010 also the higher order terms in, necessary for the three-loop calculation, are presented. In 008 the fermionic contribution [1] to a was computed and end of 00 a was completed by evaluating also the purely gluonic part which was achieved by two independent groups [8,]. In this contribution we provide some details of Ref. [8] and in particular present explicit results for the 14 master integrals containing a massless one-loop subdiagram. The results for 16 more complicated integrals have been presented in Ref. [10]. In addition one has one more finite master integral which is only known numerically and ten integrals which have no static line and are thus known since long.. Reduction to master integrals In order to compute the static potential one has to consider the four-point amplitudes describing the quark anti-quark interation. After integrating out the heavy quark mass one arrives at non-relativistic QCD and remains with only the dependence on one kinematical variable, the momentum transfer between the quarks. Consequently all occuring integrals can be mapped to one of the three integrals displayed in Fig. 1. We have performed both the direct reduction of these integrals but also applied a partial fractioning in all cases where three static lines meet in one vertex. This reduces the number of indices which have to be considered during the reduction from to twelve. 1
J1 J J J4 Figure 1. Three-loop two-point integrals with massless relativistic solid lines and static propagators zig-zag lines. J5 J6 J7 J8 i0 J J10 J11 J1 In Ref. [8] the evaluation of a has been performed for general gauge parameter ξ which is a very important check, in particular for such involved calculations as the present one. However, the computational price one has to pay is quite high: a rough estimate of the complexity based on the number of integrals which have to be reduced to masters shows that the linear ξ term is about seven times and the ξ term even 18 times more complicated than the Feynman gauge result. Let us mention that nevertheless all occurring integrals could be reduced with the help of FIRE [11].. Results for master integrals There are altogether 41 master integrals contributing to a. As already mentioned above, in this contribution we want to consider those with a massless one-loop insertion which can easily be integrated in terms of Γ functions using standard formulae. As a result one obtains the two-loop integrals in Fig. where one of the indices has a non-integer exponent involving the space-time parameter 4 d/ which is explicitly indicated next to the corresponding line. We present the analytical results for all integrals which can be expressed as one of the following functions G 1 k a1 k q a l a 1 l q a4 k l a5 v k a6 v l, a7 G k a1 k q a l a v k l a7 l q a4 k l a5 v k, a6 J1 J14 i0 Figure. Two-loop master integrals with a regularized line contributing to a. The solid and zig-zag lines correspond to relativistic and static propagators, respectively. The black box in the case of J 7 denotes a monomial in the numerator. G k a1 k q a l a v l a7 l q a4 k l a5 v k, a6 G 4 k a1 k q a l a v k l a7 l q a4 k l a5 v k, a6 G 5 k a1 k q a l a 1 l q a4 k l a5 v k a6 v l, a7 G 6 k a1 k q a l a v k l a7 l q a4 k l a5 v k a6, G 7,± k a1 k q a l a v l i0 a7 l q a4 k l a5 v k a6, where in all propagators, apart the last one, the causal i0 is implied.
In the following we set for simplicity q 1 and v 1 since the corresponding dependences can easily be reconstructed. The results for J 1 to J 14 read [ J 1 G 1 0,1,1,0,1,1,1 π 4 144 14π 5ζ 8π 101π4 60 704ζ 864 ]..., 4 7π J G 0,1,1,0,1,1,1 4π 4ζ π 56π 4π4 45 ζ 048π π4 56ζ 45 1684π 176π4 45 16π ζ 048ζ 16π ζ 160ζ5..., π6 60 68ζ 0ζ5 J G 1,1,1,1,1,1,1 8π 0ζ 16π 56π 106π4 11ζ 45 17π 688π4 18π 4 log 116ζ 45 550π ζ 1865ζ5 864s 6 660ζ5 178ζ 440π ζ 76ζ 576π Li 4 1 4π log 4 0π 4 log 7π 4 log π6 808π4 1040π..., 45 J 4 G 0,1,1,0,1,1,1 4π 8ζ π 56π 14π4 64ζ 048π 11π4 51ζ 1684π 86π4 18π ζ 1604π ζ 16ζ 40ζ5 6161π6 180 406ζ 0ζ5..., J 5 G 1,1,1,1,1,1,1 8π 40ζ 16π 56π 1π4 4ζ 17π 5 16π4 5 70ζ5 5556ζ 576π Li 4 1 6π 4 log 4ζ 144π 4 log 188π6..., 178s 6 50ζ5 5440π ζ 44π ζ 87ζ 4π log 4 84π 4 log 86π4 5 J 6 G 4 1,0,1,1,1,1,1 4π 40ζ 64π π4 160ζ 56π 8π4 640ζ 110ζ5 104π π4 560ζ 4480ζ5 6416π ζ..., 1040π 16π 1604π ζ 1400ζ 061π6 180 J 7 G 5 1,0,1,1,1,1,1 1 ζ 4 100 π 18 π4 10 1ζ 40 π π4 5 78ζ π ζ 5ζ5
4 4816 50π 44π4 10 1π6 160 1616ζ π ζ ζ 1ζ5..., J 8 G 5 1,1,1,0,1,1,1 4π 0ζ 16π 64π 16π4 80ζ 56π 64π4 0ζ 104π 6056π ζ 56π4 14π ζ 700ζ 560ζ5 671π6 180 180ζ 40ζ5..., J G 6 0,1,1,1,0,1,1 1 8 1 log log 16 log π 5 0 16π 104 log 16 log 16 4 log 104π 17π4 0 4 π log 140ζ 640 log π log 104 log 4 π log log log4 110ζ 80 logζ 716π4 87 log 11648 640π 08 π log 17 π4 log 640 log π log 08 log 4 log5 45 40 logζ 8 π log 16 log4 780ζ 80π ζ 80 log ζ 657ζ5..., J 10 G 7, 0,1,1,0,1,1,1 8π 8ζ 16 π log 64π 51π 40π4 18 π log 16 π log 64ζ 16 logζ 406π 0π4 104 π log 80 π4 log 18 π log π log 51ζ 184π ζ 18 logζ 16 log ζ 4ζ5 68π 560π4 5π6 7 81 π log 640 π4 log 104 π log 80 π4 log 56 π log 16 π log 4 406ζ 547π ζ 104 logζ 668 π logζ 18 log ζ log ζ 0ζ 1ζ5 48 logζ5..., J 11 G 7, 0,0,1,1,1,,0 1 4 log 56 log 188ζ 8 log log π 8 4 π log 8 log 1 160 8π 4 log 76 56π π4 0 log 10 16 π log 56 log 4 π log 16 log log4 75ζ
5 76 logζ 584 0π 18π4 5 log 11 π log 5 π4 log 0 log 8 π log 8 log4 564ζ 16 π log 11 log 4 log5 76π ζ 76 log ζ 75ζ5 π 16π4 5 5617π6 85 45 04 logζ 5008 11648 log 640 π log 6 5 π4 log log 11 π log 5 π4 log 640 log π log 56 log4 4 π log 4 16 log5 45 04π ζ 4 log6 0080ζ 1058 logζ 75 π logζ 04 log ζ 75 log ζ 1767ζ 064 logζ5 4..., 018ζ5 J 1 G 7, 0,1,1,0,1,1,1 4π 8ζ 8 π log π 56π 0π4 64 π log 8 π log 64ζ 16 logζ 048π 160π4 51 π log 40 π4 log 64 π log 16 π log 51ζ 0π ζ 18 logζ 16 1684π log ζ 4ζ5 180π4 18π6 6 406 π log 0 π4 log 51 π log 40 π4 log 18 π log 8 π log 4 406ζ 1160π ζ 104 logζ 040 π logζ 18 log ζ log ζ 0ζ 1ζ5 48 logζ5..., J 1 G 7, 1,1,1,1,1,1,1 8π 40ζ 16 π log 16π 56π π4 π log 16 π log 4ζ 80 logζ 17π 04π4 51 π log 404 π4 log π log π log 4ζ 41π ζ 448 logζ 80 log ζ 1176ζ5 1040π 51π4 1108π6 85 584 π log 5088 π4 log 51 π log 17864 π4 log 64 π log 16 π log 4 87ζ 86π ζ 4864 logζ 84 π logζ 448 log ζ 160 log ζ 1646ζ 864ζ5 5 logζ5...,
6 J 14 G 7, 1,1,1,1,1,1,1 16π 40ζ π log π 51π 548π4 64 π log π log 4ζ 80 logζ 584π 446π4 104 π log 5416 π4 log 64 π log 64 π log 4ζ 10544π ζ 448 logζ 80 log ζ 1176ζ5 0480π 5440π4 447π6 85 7168 π log 6 π4 log 104 π log 1876 π4 log 18 π log π log 4 87ζ 584π ζ 4864 logζ 1088 π logζ 448 log ζ 160 log ζ 1646ζ 864ζ5 5 logζ5.... In the above results the factor iπ d/ e γe is implied on the right-hand side. Furthermore, s 6 ζ{5, 1}, ζ6 1 i i 1 j i1 i 5 j1 j, and we show the expansion terms up to the order which is needed for the static potential. The results for J 1,..., J 14 have been checked numerically with the help of the program FIESTA [1,1]. 4. Result for a For completeness we repeat the result for the non-fermionic part of a, a 0, which has been obtained in Refs. [8,] a 0 50.41 C A dabcd F 16.1 d abcd A N A, where C A N c and d abcd F d abcd A /N A N c 6N c /48 in the case of SUN c. Since for three master integrals the highest expansion coefficient is only known numerically see Ref. [10] no analytical result is available yet for a 0. Note, however, that the achieved accuracy is sufficient for all foreseeable applications. Acknowledgements This work is supported by DFG through project SFB/TR Computational Particle Physics and RFBR, grant 08-0-01451. V.S. appreciates the financial support for the participation in the workshop of the Institute of Theoretical Particle Physics of KIT. REFERENCES 1. A. V. Smirnov, V. A. Smirnov and M. Steinhauser, Phys. Lett. B 668 008 [arxiv:080.1 [hep-ph]].. W. Fischler, Nucl. Phys. B 1 177 7.. A. Billoire, Phys. Lett. B 180 4. 4. M. Peter, Phys. Rev. Lett. 78 17 60 [arxiv:hep-ph/6100]. 5. M. Peter, Nucl. Phys. B 501 17 471 [arxiv:hep-ph/7045]. 6. Y. Schroder, Phys. Lett. B 447 1 1 [arxiv:hep-ph/8105]. 7. B. A. Kniehl, A. A. Penin, M. Steinhauser and V. A. Smirnov, Phys. Rev. D 65 00 00 [arxiv:hep-ph/0106]. 8. A. V. Smirnov, V. A. Smirnov and M. Steinhauser, Phys. Rev. Lett. 104 010 1100 [arxiv:011.474 [hep-ph]].. C. Anzai, Y. Kiyo and Y. Sumino, Phys. Rev. Lett. 104 010 1100 [arxiv:011.45 [hep-ph]]. 10. A. V. Smirnov, V. A. Smirnov and M. Steinhauser, arxiv:1001.668 [hep-ph]. 11. A. V. Smirnov, JHEP 0810 008 107 [arxiv:0807.4 [hep-ph]]. 1. A. V. Smirnov and M. N. Tentyukov, Comput. Phys. Commun. 180 00 75 [arxiv:0807.41 [hep-ph]]. 1. A. V. Smirnov, V. A. Smirnov and M. Tentyukov, arxiv:01.08 [hep-ph].