Three-Loop Results in HQET 3

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arXiv:0805.1474v1 [hep-ph] 10 May 2008 a Grozin A.G. HQET in results Three-loop aesaei utadul nerl n a be can and integral, arbitrary double for a calculated just easily is space nate diagrams propagator Two-loop 3. Figure Γ( rse i ucin.Tenx w nsare ones single two a next with The integrals functions. two-loop Γ via pressed [4]. Mincer is package It massless [3]. the Grinder to algo- package imple- analogous REDUCE and The a hand, as by 2). mented constructed (Fig. rela- been integrals parts has master rithm by 8 integration to can using tions, They [3], reduced 1). (Fig. be diagrams propagator HQET integra- identities. using [2] integrals, parts by master tion two to [1] duced diagrams propagator HQET Off-shell 1. J e.Tels n stuythree-loop. truly is one last The dex. 3 Γ( ntttfu hoeiceTicepyi,Universität Karlsruh Teilchenphysik, für Theoretische Institut F ( Γ( eetrslsadmtoso he-opcluain nHQ in calculations three-loop of methods and results Recent h rttolo iga Fg )i coordi- in 3) (Fig. diagram two-loop first The h rt5mse nerl a eesl ex- easily be can integrals master 5 first The three-loop of topologies generic 10 are There re- were diagrams propagator HQET Two-loop n n 2 n 1 1 1 n n , + + 1 n n 2 + 1 n n n 1 n , 2 1 2 n n + d , + 4 3 + 3 n , n a 2 + n n − n 3 3 4 3 n , 3 n , 2 2( + n n 2 + n 1 5 4 5 5 + = ) n , − n n n 2 n 4 1 d 4 2 − )Γ( + + + d n n d/ )Γ( n 3 5 3 2 + 2 − 2 + n − 4 d n n )Γ( )) 1 n n 4 ... d 4 4 − )Γ( dpnetin- -dependent 5 n − 5 [3]: d )Γ( d d/ n

2 n 1 − 1 + n . 5 n (1) ) 3 ) 1 " iepoaao nerl yivrino Eu- of inversion by momenta: integration integrals clidean mas- on-shell propagator to sive related are integrals propagator a encluae 5 sn eeburpolyno- in Gegenbauer mials using [5] calculated been has I d − (1 2 3 iial otetolo ae[] oeHQET some [7], case two-loop the to Similarly h eoddarmi i.3frarbitrary for 3 Fig. in diagram second The − Γ(2 − Taereviewed. are ET n , F n n ( 1 3 n 2 5 n 4 Γ( , n 1 n − 1 − 1 d , n − e − 1 x n 1 n d n − n , sae[6]: -space n 7 d 7 d 3)Γ(3 + − n 3 n 3)Γ(2 + = ) − 6 n n n n d − 6 3 8 n 1 n n n n 3 + 7 2)Γ 5 7 6 − n Γ n − 8 n , d n 2 n 2 d 2 n n n ( 4 + 2 − 5 n n 7 − − n n − 5 n − 2 − 4 8 1 d , Γ( d 2 2 3 n d d 3 + d 6) + n Γ d 8 − − 3) + 6) + − − = = , n n d 2 2 2 2) 5 2 n − d # − − 6 + − . n n n 2 − 8 4 d 1 6 +

1 (2) n 2 A.G. Grozin

Figure 1. Topologies of three-loop HQET propagator diagrams

3 = I1 = I1I2 = I3

2 2 I1 G1 ∼ I3 ∼ I3 I2 G2

= I1J(1, 1, −1+2ε, 1, 1) = G1I(1, 1, 1, 1,ε)

Figure 2. Master integrals for three-loop HQET propagator diagrams Three-loop results in HQET 3

n1 n1−d n1−d d − n3 − n6 − n7 − n8 Γ 2 Γ 2 + n2 + n3 Γ 2 + n2 + n4 d − n1 − n4 d − n2 − n5 2Γ(n1)Γ(n3)Γ(n4)Γ(n1 +2n2 + n3 + n4 − d) − n6 n6 n7 − n7 (3) n1 d−n1 Γ 2 + n2 + n3 + n4 − d Γ 2 − n2 n1 n3 n2 − . (5) Γ d n1 In particular, at d = 4 we have 2

= −5ζ5 + 12ζ2ζ3 . (4) Reducing the left-hand side to the master inte- Figure 4. Topologies of on-shell HQET propaga- grals, one can express [8] the last master integral tor diagrams with mass in Fig. 2 via the right-hand side of (4), which is an on-shell master integral known from [9]. Note that O(ε) corrections to (4) are not known. Using this technique, the HQET heavy-quark There are two generic topologies of three-loop on-shell HQET propagator diagrams with a mas- propagator has been calculated up to three loops [10], and the heavy-quark field anomalous sive loop (Fig. 4). Algorithms of their reduction dimension (obtained earlier by a completely dif- to master integrals, using integration by parts identities, have been constructed [13] by Gröbner ferent method [11]) has been confirmed. The anomalous dimension of the HQET heavy–light bases technique [14]. All scalar integrals can be quark current has been calculated [10]. The divided into two classes which are not mixed by correlator of two heavy–light currents has been recurrence relations: apparently even and appar- found, up to three loops, including light-quark ently odd ones. They are integrals which would be even or odd with respect to v → −v if there mass corrections of order m and m2 [12]. The quark-condensate contribution to this correlator were no i0 in denominators. has been also calculated up to three loops [12]. All apparently even integrals of the first topol- ogy reduce to Its ultraviolet divergence yields the difference of twice the anomalous dimension of the heavy- quark current and the that of the quark condensate, thus providing a completely indepen- dent confirmation of the result obtained in [10]. The gluon-condensate contribution has been calculated up to two loops [12] (at one loop it van- ishes). while apparently odd ones to 2. On-shell HQET propagator diagrams with mass 2.1. Reduction The two-loop on-shell HQET propagator integral with a massive loop can be expressed via Γ All apparently even integrals of the second topol- functions for arbitrary indices [13]: ogy reduce to

n2 n4 n3 =

n1 4 A.G. Grozin

while apparently odd ones to If p0 < 0, we can deform the integration contour

i kE0

The master integrals

ip0 −i

Γ(n − (d − 1)/2) can be easily expressed via Γ functions. The mas- 2 × In1n2 (p0)=2 1/2 ter integral π Γ(n2) ∞ d (k2 − 1)(d−1)/2−n2 cos π − n2 dk . 2 (2k − 2p )n1 Z1 0 This integral is

I (p )= has been investigated in detail [15,16]. n1n2 0 Γ(n + n − 2+ ε)Γ(n +2n − 4+2ε) 1 2 1 2 × 2.2. A master integral Γ(n2)Γ(2(n1 + n2 − 2+ ε)) Now we shall discuss the integrals n1,n1 +2n2 − 4+2ε 1 2F1 3 (1 + p0) , n n n1 + n2 − + ε 2 2 2 2

In1n2n3 = (6) n1 n1 or, after using a 2F1 identity,

n3 In1n2 (p0)= Γ(n1 + n2 − 2+ ε)Γ(n1 +2n2 − 4+2ε) (I is one of the master integrals). Several ap- × 111 Γ(n )Γ(2(n + n − 2+ ε)) proaches have been tried [13,17]. The best result 2 1 2 1 n , 1 n + n − 2+ ε was obtained [17] using a method similar to [15]. F 2 1 2 1 2 1 − p2 . (8) 2 1 n + n − 3 + ε 0 First we consider the one-loop subdiagram 1 2 2

n2 This result was obtained [17] using the HQET Feynman parametrization. Now we can integrate in dd−1~p in the three-loop In1n2 (p0)= = (7) n1 diagram: d−1 1 dk0 d ~k . Γ(n3 − 3/2+ ε) d/2 n1 2 n2 × iπ [−2(k0 + p0) − i0] [1 − k − i0] In1n2n3 = 1/2 Z π Γ(n3) +∞ d−1 After the Wick rotation, we integrate in d ~k: 2 2 3/2−n3−ε In1n2 (ipE0)(1 + pE0) dpE0 . (9) −∞ Γ(n − (d − 1)/2) Z I (p )= 2 × n1n2 0 1/2 The square of 2F1 in (8) can be expressed via π Γ(n2) an F using the Clausen identity. We analyt- +∞ 2 (d−1)/2−n2 3 2 (kE0 + 1) 2 dk . ically continue this 3F2 from 1 + pE0 > 1 to E0 n1 −∞ (−2p0 − 2ikE0) 2 Z z = 1/(1 + pE0) < 1 and integrate (9) term by Three-loop results in HQET 5

term. The result contains, in general, three 4F3 2.3. Other master integrals of unit argument. Now we turn to other master integrals [13]. The convergent integral I122 is related to the By applying Mellin–Barnes techniques to the α- master integral I111 by representation and using the Barnes lemmas, we succeeded in calculating one integral exactly: (d − 3)2(d − 4)(3d − 8)(3d − 10) I = − I 122 8(3d − 11)(3d − 13) 111 = For this integral, we obtain [17] Γ(1/2 − ε)Γ(−ε)Γ2(2ε)Γ(1 + ε)Γ(3ε − 1) I 1 1 4Γ(3/2 − ε)Γ(4ε) 122 = − 3 2 [ψ(1/2 − ε)+ ψ(1 − ε) − 2 log 2 + 2γ ] . (13) Γ (1 + ε) 2ε "1+2ε E 1, 1 − ε, 1+ ε, −2ε F 2 1 In other master integral calculations, we use 4 3 3 + ε, 1 − ε, 1 − 2ε 2 the Mellin–Barnes of the massive two-point func- 2 Γ2(1 − ε)Γ3(1+2 ε) tion: − 1+4ε Γ2(1 + ε)Γ(1 − 2ε)Γ(1 + 4ε) 1 n1 2 , 1+2ε, −ε 3F2 1 (10) 3 +2ε, 1 − ε = 2 1 + n2 1+6ε 1 ddk 2 − 4 − 2 Γ (1 ε)Γ (1+2ε)Γ(1 2ε)Γ (1+3ε) d/2 2 2 n1 2 2 n2 4 . iπ [m − k − i0] [m − (k + p) − i0] Γ (1 + ε)Γ(1 + 4ε)Γ(1 − 4ε)Γ(1 + 6ε) # Z− ∞ md 2(n1+n2) 1 +i = dz Γ(−z)× 7 Γ(n )Γ(n ) 2πi − ∞ Expansion of this result up to ε agrees with [17] 1 2 Z i Γ(n1 + z)Γ(n2 + z)Γ(n1 + n2 − d/2+ z) 2 3 2 × I122 π Γ (1+2ε)Γ (1+3ε) Γ(n1 + n2 +2z) 3 = 6 . (11) Γ (1 + ε) 3 Γ (1 + ε)Γ(2 + 6ε) − −z m 2z (14) This equality has also been checked by high pre- cision numerical calculations at some ﬁnite ε val- Applying this representation and eliminating ues. During the workshop, David Broadhurst has one of the lines in the triangle with unit indices rewritten this conjectured hypergeometric iden- using integration by parts, we obtain a single tity in a nice form Mellin–Barnes integral: Γ(1 − ε)Γ(1 + 2ε) b(ε)= , Γ2(2ε)Γ(3ε − 1) Γ(1 + ε) = × 4Γ(4ε) bn(ε) ∞ gn(ε)= , 1 +i b(nε)(1 + 2nε) dz Γ(−z)Γ(−2ε − z)Γ(−ε − z)× 1, 1 − ε, 1+ ε, −2ε 2πi −i∞ g (ε) F 2 1 Z 1 4 3 3 + ε, 1 − ε, 1 − 2ε Γ(1 + z)Γ(1/2+ ε + z)Γ(1 + ε + z) 2 = 1 , 1+2ε, − ε Γ(1 − 2ε − z)Γ(3/2+ ε + z) − 2g (ε) F 2 1 2 3 2 3 +2ε, 1 − ε π2 6ζ − 5π2 2 −Γ3(1 + ε) − 3 2 + g3(ε)=0 . (12) " 9ε 9ε

11 4 10 19 2 We have no analytical proof. + π − ζ3 + π 270 3 9 6 A.G. Grozin

Figure 5. Topologies of three-loop massive on-shell propagator diagrams

8 8 2 11 4 38 65 2 integrals of this type. Two extra terms of ε ex- + − ζ5 + π ζ3 + π − ζ3 + π ε 3 9 54 3 9 pansion of the master integral of Sect. 2.2 were required for this calculation which were not ob- + ··· . (15) tained in [13]. This was the initial motivation # for [17]. Similarly, in this case, we apply (14) once and use (5): 3. On-shell massive QCD propagator diagrams π3/2 = × These diagrams are used for calculation of 4εΓ(3/2 − ε) QCD/HQET matching coefficients [19]. There +i∞ 1 Γ(1 + z)Γ(3/2 − ε + z)Γ(ε + z) are two generic topologies of two-loop on-shell dz × propagator diagrams. They were reduced [20,15, 2πi Γ(3/2+ z)Γ(ε − z) −Zi∞ 21] to three master integrals, using integration by Γ(−z)Γ(−1/2+ ε − z)Γ(−3/2+2ε − z) parts. There are 11 generic topologies of three-loop 32 = − π2 + ··· (16) HQET propagator diagrams (Fig. 5) (10 of them 3 are the same as in HQET, Fig. 1, plus the dia- Feynman integrals considered here were gram with a closed heavy-quark loop which does used [13] for calculating the matching coeffi- not exist in HQET). They can be reduced [11] cients for the HQET heavy-quark field and the to 19 master integrals (Fig. 6) using integration heavy–light quark current between the b-quark by parts. The reduction algorithm was first im- HQET with dynamic c-quark loops and without plemented as a FORM package SHELL3 [11]; in such loops (the later theory is the low-energy some papers [22], reduction is performed using approximation for the former one at scales below the Laporta method. The master integrals are mc). Another recent application — the effect known from [9,11] (see also [22]). of mc =6 0 on b → c plus lepton pair at three As an example, let’s consider chromomagnetic loops [18]. The method of regions was used; the interaction of the heavy quark in HQET. The purely soft region (loop momenta ∼ mc) gives two-loop anomalous dimension of the chromo- Three-loop results in HQET 7

Figure 6. Master integrals for three-loop massive on-shell propagator diagrams 8 A.G. Grozin magnetic interaction operator has been calculated (2002) 011502. in [23], using off-shell HQET diagrams (Sect. 1) 9. S. Laporta, E. Remiddi, Phys. Lett. B 379 and R∗ operation to get rid of infrared diver- (1996) 283. gences. It was confirmed [24], a week later, using 10. K.G. Chetyrkin, A.G. Grozin, Nucl. Phys. B a completely different approach — QCD/HQET 666 (2003) 289. matching. In the same paper, also the coeffi- 11. K. Melnikov, T. van Ritbergen, Nucl. Phys. cient of the chromomagnetic interaction opera- B 591 (2000) 515. tor in the HQET Lagrangian has been calculated 12. K.G. Chetyrkin, A.G. Grozin, unpublished. up to two loops. Recently, both the anomalous 13. A.G. Grozin, A.V. Smirnov, V.A. Smirnov, dimension and the interaction coefficient have JHEP 11 (2006) 022. been calculated at three loops [25], also using 14. A.V. Smirnov, V.A. Smirnov, JHEP 01 the QCD/HQET matching. These results con- (2006) 001; tain two colour factors which don’t reduce to CF A.V. Smirnov, JHEP 04 (2006) 026. and CA (due to gluonic “light-by-light” diagrams; 15. D.J. Broadhurst, Z. Phys. C 54 (1992) 599. in the β-function and basic anomalous dimensions 16. D.J. Broadhurst, hep-th/9604128. of QCD, these colour factors first appear at four 17. A.G. Grozin, T. Huber, D. Maˆıtre, JHEP 07 loops. (2007) 033. I am grateful to S. Bekavac, D.J. Broadhurst, 18. A. Czarnecki, A. Pak, arXiv:0803.0960 [hep- K.G. Chetyrkin, A. Czarnecki, A.I. Davydychev, ph]. T. Huber, D. Maˆıtre, P. Marquard, J.H. Piclum, 19. D.J. Broadhurst, A.G. Grozin, Phys. Rev. D D. Seidel, A.V. Smirnov, V.A. Smirnov, M. Stein- 52 (1995) 4082. hauser for collaboration on multi-loop projects in 20. N. Gray, D.J. Broadhurst, W. Grafe, HQET; to the organizers of Loops and Legs 2008; K. Schilcher, Z. Phys. C 48 (1990) 673; and to D.J. Broadhurst for discussions of the mas- D.J. Broadhurst, N. Gray, K. Schilcher, Z. ter integral of Sect. 2.2. Phys. C 52 (1991) 111. 21. J. Fleischer, O.V. Tarasov, Phys. Lett. B 283 REFERENCES (1992) 129; Comput. Phys. Commun. 71 (1992) 193. 1. D.J. Broadhurst, A.G. Grozin, Phys. Lett. B 22. P. Marquard, L. Mihaila, J.H. Piclum, 267 (1991) 105. M. Steinhauser, Nucl. Phys. B 773 (2007) 1. 2. K.G. Chetyrkin, F.V. Tkachov, Nucl. Phys. 23. G. Amorós, M. Beneke, M. Neubert, Phys. B 192 (1981) 159. Lett. B 401 (1997) 81. 3. A.G. Grozin, JHEP 03 (2000) 013. 24. A. Czarnecki, A.G. Grozin, Phys. Lett. B 405 4. S.G. Gorishny, S.A. Larin, F.V. Tkachov, (1997) 142; Erratum B 650 (2007) 447. Preprint INR P-0330, Moscow (1984); 25. A.G. Grozin, P. Marquard, J.H. Piclum, S.G. Gorishny, S.A. Larin, L.R. Surguladze, M. Steinhauser, Nucl. Phys. B 789 (2008) F.V. Tkachov, Comput. Phys. Commun. 55 277. (1989) 381; S.A. Larin, F.V. Tkachov, J.A.M. Ver- maseren, preprint NIKHEF-H/91-18, Ams- terdam (1991). 5. M. Beneke, V.M. Braun, Nucl. Phys. B 426 (1994) 301. 6. K.G. Chetyrkin, A.L. Kataev, F.V. Tkachov, Nucl. Phys. B 174 (1980) 345. 7. D.J. Broadhurst, A.G. Grozin, hep-ph/9504400. 8. A. Czarnecki, K. Melnikov, Phys. Rev. D 66