Relation Between the Pole and the Minimally Subtracted Mass in Dimensional Regularization and Dimensional Reduction to Three-Loop Order

Home , Pole mass

arXiv:hep-ph/0702185v1 19 Feb 2007 hep-ph/0702185 DESY07-016 TTP07-04 SFB/CPP-07-04 utatdms ndmninlrglrzto and regularization dimensional in mass subtracted ASnmes 23.t1.5- 46.y14.65.Ha 14.65.Fy 14.65.-q 12.38.-t numbers: PACS re analytical the of check ago. independent years first the provide thus nselrnraiainconstants renormalization on-shell euaiainshm.Seilepai spto h evan the on dimensio put the applying is of emphasis QCD renormalization Special of the framework scheme. the regularization in a mass quark tracted eainbtentepl n h minimally the and pole the between Relation ecmueterlto ewe h oeqakms n h mi the and mass quark pole the between relation the compute We a ntttfu hoeiceTicepyi,Universit Teilchenphysik, für Theoretische Institut (a) iesoa euto otrelo order three-loop to reduction dimensional .Marquard P. b I ntttfu hoeicePyi,Universität Ha Physik, für Theoretische Institut II. (b) ( a ) .Mihaila L. , ε saa as sab-rdc eoti h three-loop the obtain we by-product a As mass. -scalar 62 alrh,Germany Karlsruhe, 76128 26 abr,Germany Hamburg, 22761 ( a Z ) ..Piclum J.H. , Abstract m OS and Z 2 OS ndmninlrglrzto and regularization dimensional in ( a,b ) n .Steinhauser M. and a alrh (TH) ät Karlsruhe ut optdseveral computed sults setculnsand couplings escent mburg a euto as reduction nal ial sub- nimally ( a ) 1 Introduction

In quantum chromodynamics (QCD), like in any other renormalizeable quantum field theory, it is crucial to specify the precise meaning of the parameters appearing in the underlying Lagrangian — in particular when higher order quantum corrections are considered. The canonical choice for the coupling constant of QCD, αs, is the so-called modified minimal subtraction (MS) scheme [1] which has the advantage that the beta function, ruling the scale dependence of the coupling, is mass-independent. On the other hand, for a heavy quark besides the MS scheme also other definitions are important — first and foremost the pole mass. Whereas the former definition is appropriate for those processes where the relevant energy scales are much larger than the quark mass the pole mass is the relevant definition for threshold processes. Thus it is important to have precise conversion formulae at hand in order to convert one definition into the other. By far the most loop calculations performed within QCD are based on dimensional regularization (DREG) in order to handle the infinities which occur in intermediate steps. It is well known, however, that it is not convenient to apply DREG to supersymmetric theories since it introduces a mismatch between the numbers of fermionic and bosonic degrees of freedom in super-multiplets. In order to circumvent this problem and, at the same time, take over as many advantages as possible from DREG the regularization scheme dimensional reduction (DRED) has been invented (see, e.g., Refs. [2,3]). Indeed, the application of DRED to supersymmetric theories leads to a relatively small price one has to pay at the technical level. An elegant way is to introduce an additional scalar particle (the so-called ε scalar) at the level of the Lagrangian and to proceed for the practical calculation of the Feynman diagrams as in DREG. The situation becomes more complicated in case the symmetry between fermions and bosons in the underlying theory is distorted, e.g., after some heavy squarks have been integrated out. In such situations couplings involving the ε scalar, which are called evanescent couplings, renormalize differently from the gauge couplings and therefore one has to allow for new couplings in the theory. The same is true if DRED is applied to QCD: in addition to αs four new couplings have to be introduced, each of which has their own beta function governing both the running and the renormalization. In this paper we take over the notation from [4] and denote them by αe (the coupling between ε scalars and quarks) and ηi (i = 1, 2, 3), which describe three different four-ε vertices. The one-loop relation between the MS and pole mass has been considered long ago in Ref. [5]. At the beginning of the nineties the mass relation to two-loop order [6] was one of the first applications of two-loop on-shell integrals. In a subsequent paper also the two-loop result for the on-shell wave function counterterm has been obtained [7]. The three-loop mass relation has been computed for the first time in Ref. [8,9] where the off- shell fermion propagator has been considered for small and large external momenta. The on-shell quantities have been obtained with the help of a conformal mapping and Padé approximation. The numerical results of Ref. [8,9] have later been confirmed in Ref. [10] OS by an analytical on-shell calculation. The three-loop result for Z2 has been obtained in

2 Ref. [11]. In this paper we provide the first independent check of the analytical results OS OS for Zm and Z2 in the DREG scheme. The renormalization scheme based on DRED together with modified minimal subtraction is called DR. As far as the relation between the DR and the pole mass is concerned one can find the one-loop result in Ref. [12]. The two-loop calculation [13] has been performed in DRED, identifying, however, the evanescent coupling αe with αs. In this paper we will provide the more general result and furthermore extend the relation to three-loop order. The paper is organized as follows: In Section 2 we outline the general strategy, provide technical details and derive the result for the on-shell mass and wave function counterterm in the case of dimensional regularization. The peculiarities of dimensional reduction are explained in Section 3 where all relevant counterterms are listed. Finally, Section 4 con- tains the main result of our paper: the relation between the pole mass and the minimally subtracted mass in the framework of dimensional reduction up to three-loop order.

2 On-shell counterterms for mass and wave function in dimensional regularization

In order to compute the counterterms for the quark mass and wave function one has to put certain requirements on the pole and the residual of the quark propagator. More precisely, in the on-shell scheme we demand that the quark two-point function has a zero at the position of the on-shell mass and that the residual of the propagator is −i. Thus, the renormalized quark propagator is given by

OS −iZ2 SF (q) = (1) q/ − mq,0 + Σ(q, Mq) 2→ 2 q Mq −i −→ , (2) q/ − Mq where the renormalization constants are deﬁned as

OS mq,0 = Zm Mq , (3)

OS ψ0 = Z2 ψ . (4) q ψ is the quark ﬁeld with mass mq, Mq is the on-shell mass and bare quantities are denoted by a subscript 0. Σ denotes the quark self-energy contributions which can be decomposed as

2 2 Σ(q, mq) = mq Σ1(q , mq)+(q/ − mq) Σ2(q , mq) . (5)

OS OS The calculation outlined in Ref. [6] for the evaluation of Zm and Z2 reduces all occurring Feynman diagrams to the evaluation of on-shell integrals at the bare mass scale. In

3 particular, it avoids the introduction of explicit counterterm diagrams. At three-loop order we ﬁnd it more convenient to follow the more direct approach described in Ref. [10, 11], which requires the calculation of diagrams with mass counterterm insertion. Following 2 2 the latter reference, we expand Σ around q = Mq

2 2 Σ(q, Mq) ≈ Mq Σ1(Mq , Mq)+(q/ − Mq) Σ2(Mq , Mq) ∂ +M Σ (q2, M ) (q2 − M 2)+ . . . q 2 1 q 2 2 q ∂q q =Mq

≈ M Σ (M 2, M ) q 1 q q ∂ +(q/ − M ) 2M 2 Σ (q2, M ) + Σ (M 2, M ) + ... . (6) q q 2 1 q 2 2 2 q q ∂q q =Mq

Inserting Eq. (6) into Eq. (1) and comparing to Eq. (2) we ﬁnd the following formulae for the renormalization constants

OS 2 Zm = 1+Σ1(Mq , Mq) , (7) −1 ∂ ZOS = 1+2M 2 Σ (q2, M ) + Σ (M 2, M ) . (8) 2 q 2 1 q 2 2 2 q q ∂q q =Mq

2 2 If we consider the external momentum of the quarks to be q = Q(1 + t) with Q = Mq , the self-energy can be written as

2 2 2 Σ(q, Mq)= MqΣ1(q , Mq)+(Q/ − Mq)Σ2(q , Mq)+ tQ/ Σ2(q , Mq) . (9)

Q/ +Mq Let us now consider the quantity Tr{ 2 Σ} and expand to ﬁrst order in t 4Mq

Q/ + M Tr q Σ(q, M ) = Σ (q2, M )+ tΣ (q2, M ) 4M 2 q 1 q 2 q q ∂ = Σ (M 2, M )+ 2M 2 Σ (q2, M ) +Σ (M 2, M ) t 1 q q q 2 1 q 2 2 2 q q ∂q q =Mq +O t2 . (10)

OS 2 2 OS Thus, to obtain Zm one only needs to calculate Σ1 for q = Mq . To calculate Z2 , one has to compute the ﬁrst derivative of the self-energy diagrams. The mass renormalization is taken into account iteratively by calculating one- and two-loop diagrams with zero- momentum insertions. Sample diagrams contributing to the quark propagator are shown in Fig. 1. All occurring (3) (3) Feynman integrals can be mapped onto J+ and L+ as given in Eq. (4) of Ref. [14] and to seven more similar ones which will be listed in Ref. [15]. All Feynman diagrams are generated with QGRAF [16] and the various topologies are iden- tiﬁed with the help of q2e and exp [17, 18]. In a next step the reduction of the various functions to so-called master integrals has to be achieved. For this step we use the so-called

4 (a)(b)(c)

(d)(e)(f)

Figure 1: Sample three-loop diagrams. Solid lines denote massive quarks with mass mq and curly lines denote gluons. In the closed fermion loops all quark ﬂavours have to be considered.

(a) (b) (c)

Figure 2: Three-loop master integrals. Solid lines denote massive and dashed lines are massless scalar propagators.

Laporta method [19,20] which reduces the three-loop integrals to 19 master integrals. We use the implementation of Laporta’s algorithm in the program Crusher [21]. It is written in C++ and uses GiNaC [22] for simple manipulations like taking derivatives of polynomial quantities. In the practical implementation of the Laporta algorithm one of the most time-consuming operations is the simplification of the coefficients appearing in front of the individual integrals. This task is performed with the help of Fermat [23] where a special interface has been used (see Ref. [24]). The main features of the implementation are the automated generation of the integration-by-parts (IBP) identities [25] and a complete symmetrization of the diagrams. The results for the master integrals can be found in Ref. [11]. We have checked the results numerically with the Mellin-Barnes method [26,27] and the program MB [28]. We did, however, find some differences to Ref. [11]: In addition to the 18 master integrals given in that reference, we found the integral depicted in Fig. 2(a) where the result can be found in Eq. (A.2) of Ref. [14]. Furthermore, we already pointed out in Ref. [14], that the result for Fig 2(b) (denoted by I16 in Ref. [11]) is wrong. As it turns out, the result given in [11] corresponds to the integral depicted in Fig. 2(c). Since there is a one-to-one correspondence between the two integrals with regard to the IBP relations, it does not matter which is chosen to be the master integral.

5 Most of the master integrals of Ref. [11] were already calculated in Ref. [19]. However, it turns out that there are diﬀerences in the order O (ǫ) terms of the integrals I2–I7 of these references. The expressions given in Ref. [11] agree with our numerical results. The results for the renormalization constants can be cast into the following form

−ǫ 2 −2ǫ α (µ) eγE α (µ) eγE ZOS = 1+ s δZ(1) + s δZ(2) i π 4π i π 4π i 3 −3ǫ α (µ) eγE + s δZ(3) + O α4 , (11) π 4π i s with i ∈ {m, 2}. It is convenient to further decompose the three-loop contribution in terms of the diﬀerent colour factors

(3) 3 FFF 2 FFA 2 FAA δZi = CF Zi + CF CA Zi + CF CA Zi FFL FAL FLL FHL +CF TF nl CF Zi + CA Zi + TF nl Zi + TF nh Zi FFH FAH FHH +CF TF nh CF Zi + CA Zi + TF nh Zi , (12) 2 where CF = (Nc − 1)/(2Nc) and CA = Nc are the eigenvalues of the quadratic Casimir operators of the fundamental and adjoint representation of SU(Nc), respectively. In the case of QCD we have Nc = 3. TF = 1/2 is the index of the fundamental representation and nf = nl + nh is the number of quark flavours. nl and nh are the number of light and heavy quark flavours, respectively. The former are considered to be massless, while the latter have mass mq. Although we have nh = 1 in our applications, we keep a generic label which is useful when tracing the origin of the individual contributions. αs(µ) is the strong coupling constant defined in the MS scheme with nf active flavours. The one- and two-loop contributions to the mass renormalization constant are

3 3 1 3 δZ(1) = −C +1+ L + 2+ π2 + L + L2 ǫ m F 4ǫ 4 µ 16 µ 8 µ 1 1 1 1 1 + 4+ π2 − ζ + 2+ π2 L + L2 + L3 ǫ2 , (13) 12 4 3 16 µ 2 µ 8 µ

6 and 9 45 9 1 199 17 1 3 45 9 δZ(2) = + + L + − π2 + π2 ln 2 − ζ + L + L2 m 32ǫ2 64 16 µ ǫ 128 64 2 4 3 32 µ 16 µ 677 205 135 7 1 + − π2 +3π2 ln 2 − π2 ln2 2 − ζ + π4 − ln4 2 − 12a 256 128 16 3 40 2 4 199 17 3 45 3 + − π2 + π2 ln 2 − ζ L + L2 + L3 ǫ C2 64 32 2 3 µ 32 µ 8 µ F 11 97 1111 1 1 3 185 11 + − − + π2 − π2 ln2+ ζ − L − L2 32ǫ2 192ǫ 384 12 4 8 3 96 µ 32 µ 8581 271 3 1 13 7 1 + − + π2 − π2 ln2+ π2 ln2 2+ ζ − π4 + ln4 2+6a 768 1152 2 2 4 3 80 4 4 1463 7 1 3 229 11 + − + π2 − π2 ln2+ ζ L − L2 − L3 ǫ C C 192 64 2 4 3 µ 96 µ 32 µ A F 1 5 71 1 13 1 + − + + + π2 + L + L2 8ǫ2 48ǫ 96 12 24 µ 8 µ 581 97 103 3 17 1 + + π2 + ζ + + π2 L + L2 + L3 ǫ C T n 192 288 3 48 16 µ 24 µ 8 µ F F l 1 5 143 1 13 1 1133 227 + − + + − π2 + L + L2 + − π2 8ǫ2 48ǫ 96 6 24 µ 8 µ 192 288 7 175 5 17 1 +π2 ln 2 − ζ + − π2 L + L2 + L3 ǫ C T n , (14) 2 3 48 16 µ 24 µ 8 µ F F h 2 2 where Lµ = ln(µ /Mq ). ζn is Riemann’s zeta function with integer argument n and a4 = 2 Li4(1/2). We give the one- and two-loop results to order O (ǫ ) and O (ǫ), respectively. In general these terms are necessary for three-loop calculations. The individual three-loop terms read

9 63 27 1 457 111 3 ZFFF = − − + L + − + π2 − π2 ln 2 m 128ǫ3 256 128 µ ǫ2 512 512 8 9 189 81 1 14225 6037 + ζ − L − L2 − − π2 +5π2 ln 2 16 3 256 µ 256 µ ǫ 3072 3072 5 153 73 1 5 1 + π2 ln2 2+ ζ − π4 − π2ζ + ζ − ln4 2 − 3a 4 128 3 480 16 3 8 5 8 4 1371 333 9 27 567 81 + − + π2 − π2 ln2+ ζ L − L2 − L3 , (15) 512 512 8 16 3 µ 512 µ 256 µ

7 33 49 33 1 3311 43 3 ZFFA = − + − L + − π2 + π2 ln 2 m 128ǫ3 768 128 µ ǫ2 1536 512 16 9 379 33 1 100247 6545 25 − ζ + L + L2 + + π2 + π2 ln 2 32 3 256 µ 256 µ ǫ 9216 9216 36 89 3995 1867 19 45 35 35 − π2 ln2 2 − ζ + π4 − π2ζ + ζ − ln4 2 − a 72 384 3 8640 16 3 16 5 144 6 4 14311 1135 71 71 1797 121 + − π2 + π2 ln 2 − ζ L + L2 + L3 , (16) 1536 1536 48 32 3 µ 512 µ 256 µ

121 1679 11413 1322545 1955 115 ZFAA = − + − − − π2 − π2 ln 2 m 576ǫ3 3456ǫ2 20736ǫ 124416 3456 72 11 1343 179 51 65 11 11 + π2 ln2 2+ ζ − π4 + π2ζ − ζ + ln4 2+ a 36 288 3 3456 64 3 32 5 72 3 4 13243 11 11 11 2341 121 + − + π2 − π2 ln2+ ζ L − L2 − L3 , (17) 1728 72 24 16 3 µ 1152 µ 576 µ

3 5 3 1 65 7 1 23 3 1 ZFFL = + − + L − + π2 + ζ + L + L2 m 32ǫ3 192 32 µ ǫ2 384 128 4 3 64 µ 64 µ ǫ 575 1091 11 2 145 119 1 + + π2 − π2 ln2+ π2 ln2 2+ ζ − π4 + ln4 2 2304 2304 9 9 96 3 2160 9 8 497 5 1 1 117 11 + a − − π2 + π2 ln2+ ζ L − L2 − L3 , (18) 3 4 384 384 3 4 3 µ 128 µ 64 µ

11 121 139 1 1 70763 175 11 ZFAL = − + + ζ + + π2 + π2 ln 2 m 72ǫ3 432ǫ2 1296 4 3 ǫ 15552 432 18 1 89 19 1 4 − π2 ln2 2+ ζ + π4 − ln4 2 − a 9 144 3 2160 18 3 4 869 7 1 1 373 11 + + π2 + π2 ln2+ ζ L + L2 + L3 , (19) 216 72 6 2 3 µ 288 µ 72 µ

1 5 35 2353 13 7 ZFLL = − + + − − π2 − ζ m 36ǫ3 216ǫ2 1296ǫ 7776 108 18 3 89 1 13 1 − + π2 L − L2 − L3 , (20) 216 18 µ 72 µ 36 µ

1 5 35 5917 13 2 ZFHL = − + + − + π2 + ζ m 18ǫ3 108ǫ2 648ǫ 3888 108 9 3 143 1 13 1 − − π2 L − L2 − L3 , (21) 108 18 µ 36 µ 18 µ

8 3 5 3 1 281 17 1 23 3 1 ZFFH = + − + L − − π2 + ζ + L + L2 m 32ǫ3 192 32 µ ǫ2 384 128 4 3 64 µ 64 µ ǫ 5257 1327 5 1 37 91 1 − − π2 + π2 ln 2 − π2 ln2 2+ ζ + π4 + ln4 2 2304 6912 36 9 96 3 2160 9 8 1145 221 1 1 117 11 + a − − π2 + π2 ln2+ ζ L − L2 − L3 , (22) 3 4 384 384 3 4 3 µ 128 µ 64 µ 11 121 139 1 1 144959 449 32 ZFAH = − + + ζ + − π2 + π2 ln 2 m 72ǫ3 432ǫ2 1296 4 3 ǫ 15552 144 9 1 109 43 1 5 1 4 + π2 ln2 2 − ζ − π4 + π2ζ − ζ − ln4 2 − a 18 144 3 1080 8 3 8 5 18 3 4 583 13 1 1 373 11 + − π2 + π2 ln2+ ζ L + L2 + L3 , (23) 108 36 6 2 3 µ 288 µ 72 µ 1 5 35 9481 4 11 ZFHH = − + + − + π2 + ζ m 36ǫ3 216ǫ2 1296ǫ 7776 135 18 3 197 1 13 1 − − π2 L − L2 − L3 . (24) 216 9 µ 72 µ 36 µ (1) For the wave function renormalization constant, we have at the one-loop level δZ2 = (1) δZm . The two-loop contribution to the wave function renormalization constant reads 9 51 9 1 433 49 3 51 9 δZ(2) = + + L + − π2 + π2 ln 2 − ζ + L + L2 2 32ǫ2 64 16 µ ǫ 128 64 2 3 32 µ 16 µ 211 339 23 297 7 + − π2 + π2 ln 2 − 2π2 ln2 2 − ζ + π4 − ln4 2 − 24a 256 128 4 16 3 20 4 433 49 51 3 + − π2 +2π2 ln 2 − 3ζ L + L2 + L3 ǫ C2 64 32 3 µ 32 µ 8 µ F 11 127 1705 5 1 3 215 11 + − − + π2 − π2 ln2+ ζ − L − L2 32ǫ2 192ǫ 384 16 2 4 3 96 µ 32 µ 9907 769 23 129 7 1 + − + π2 − π2 ln2+ π2 ln2 2+ ζ − π4 + ln4 2+12a 768 1152 8 16 3 40 2 4 2057 109 3 259 11 + − + π2 − π2 ln2+ ζ L − L2 − L3 ǫ C C 192 192 2 3 µ 96 µ 32 µ A F 1 11 113 1 19 1 + − + + + π2 + L + L2 8ǫ2 48ǫ 96 12 24 µ 8 µ 851 127 145 3 23 1 + + π2 + ζ + + π2 L + L2 + L3 ǫ C T n 192 288 3 48 16 µ 24 µ 8 µ F F l 1 1 1 947 5 11 3 17971 445 + + L + − π2 + L + L2 + − π2 16 4 µ ǫ 288 16 24 µ 8 µ 1728 288 85 1043 29 5 7 +2π2 ln 2 − ζ + − π2 L + L2 + L3 ǫ C T n . (25) 12 3 144 48 µ 8 µ 24 µ F F h 9 Again, we give the result to order O (ǫ), which is necessary for three-loop calculations. The individual three-loop terms are given by 9 81 27 1 1039 303 3 ZFFF = − − + L + − + π2 − π2 ln 2 2 128ǫ3 256 128 µ ǫ2 512 512 4 9 243 81 1 10823 58321 685 + ζ − L − L2 − − π2 + π2 ln 2 8 3 256 µ 256 µ ǫ 3072 9216 48 739 41 1 5 5 +3π2 ln2 2 − ζ − π4 + π2ζ − ζ − ln4 2 − 10a 128 3 120 8 3 16 5 12 4 3117 909 9 27 729 81 + − + π2 − π2 ln2+ ζ L − L2 − L3 , (26) 512 512 4 8 3 µ 512 µ 256 µ

33 95 33 1 1787 131 3 ZFFA = − + − L + − π2 + π2 ln 2 2 128ǫ3 768 128 µ ǫ2 512 512 8 5 469 33 1 136945 29695 755 − ζ + L + L2 + + π2 − π2 ln 2 8 3 256 µ 256 µ ǫ 9216 9216 288 235 6913 1793 45 145 55 55 − π2 ln2 2 − ζ + π4 − π2ζ + ζ − ln4 2 − a 72 384 3 3456 16 3 16 5 144 6 4 25609 3335 71 37 2155 121 + − π2 + π2 ln 2 − ζ L + L2 + L3 , (27) 1536 1536 24 8 3 µ 512 µ 256 µ

121 2009 12793 3 1 1 3 ZFAA = − + − + ζ + π4 + + ζ 2 576ǫ3 3456ǫ2 20736 128 3 1080 768 256 3 1 1 1654711 4339 325 127 − π4 ξ − − π2 − π2 ln2+ π2 ln2 2 4320 ǫ 124416 3456 144 144 5857 3419 127 37 85 85 + ζ − π4 + π2ζ − ζ + ln4 2+ a 576 3 23040 72 3 6 5 288 12 4 13 1 13 17 1 7 + − − π2 − ζ + π4 + π2ζ + ζ ξ 768 256 256 3 27648 144 3 384 5 36977 55 11 167 1 + − + π2 − π2 ln2+ ζ − π4 3456 96 12 128 3 360 1 9 1 2671 121 + − − ζ + π4 ξ L − L2 − L3 , (28) 256 256 3 1440 µ 1152 µ 576 µ

3 19 3 1 235 7 1 41 3 1 ZFFL = + − + L − + π2 + ζ + L + L2 2 32ǫ3 192 32 µ ǫ2 384 128 4 3 64 µ 64 µ ǫ 3083 2845 47 4 473 229 2 − + π2 − π2 ln2+ π2 ln2 2+ ζ − π4 + ln4 2 2304 2304 18 9 96 3 2160 9 16 1475 133 2 1 179 11 + a + − + π2 − π2 ln2+ ζ L − L2 − L3 , (29) 3 4 384 384 3 4 3 µ 128 µ 64 µ 10 11 169 313 1 1 111791 13 47 ZFAL = − + + ζ + + π2 + π2 ln 2 2 72ǫ3 432ǫ2 1296 4 3 ǫ 15552 48 36 2 35 19 1 8 − π2 ln2 2 − ζ + π4 − ln4 2 − a 9 72 3 1080 9 3 4 169 1 1 1 469 11 + − π2 + π2 ln2+ ζ L + L2 + L3 , (30) 27 18 3 4 3 µ 288 µ 72 µ 1 11 5 5767 19 7 ZFLL = − + + − − π2 − ζ 2 36ǫ3 216ǫ2 1296ǫ 7776 108 18 3 167 1 19 1 − + π2 L − L2 − L3 , (31) 216 18 µ 72 µ 36 µ 1 1 1 5 1 1 1 1 ZFHL = + L + − + π2 − L + L2 2 36 12 µ ǫ2 216 144 9 µ 24 µ ǫ 4721 19 1 329 25 7 5 − + π2 − ζ + − + π2 L − L2 − L3 , (32) 1296 54 36 3 108 144 µ 12 µ 72 µ 7 3 1 707 15 29 15 1 ZFFH = − + L − − π2 + L + L2 2 192 16 µ ǫ2 384 64 64 µ 32 µ ǫ 76897 11551 7 1 1763 31 1 − − π2 + π2 ln 2 − π2 ln2 2+ ζ + π4 + ln4 2 6912 20736 18 2 288 3 720 2 2891 233 2 143 19 +12a + − + π2 − π2 ln2+ ζ L − L2 − L3 , (33) 4 384 192 3 3 µ 128 µ 32 µ 1 − ξ 7 1 41 1 1 13 41 ZFAH = − − ξ + + ξ L + − π2 2 192ǫ3 72 64 192 64 µ ǫ2 216 2304 35 1 83 3 35 3 1 − + π2 ξ + + ξ L − + ξ L2 576 768 144 64 µ 384 128 µ ǫ 49901 36019 80 1 77 17 11 + − π2 + π2 ln2+ π2 ln2 2 − ζ − π4 + π2ζ 2592 5184 9 3 16 3 360 48 3 15 1 407 1 7 − ζ − ln4 2 − 8a + + π2 − ζ ξ 16 5 3 4 1728 256 192 3 4141 641 1 1 35 1 + − π2 + π2 ln 2 − ζ − + π2 ξ L 432 768 3 2 3 192 256 µ 35 9 247 3 + + ξ L2 + − ξ L3 , (34) 16 128 µ 1152 128 µ 1 5 1 1 8425 2 7 ZFHH = − + L2 − + π2 + ζ 2 72ǫ2 432 12 µ ǫ 2592 45 3 3 481 5 11 1 − − π2 L − L2 − L3 . (35) 216 24 µ 72 µ 6 µ 11 Starting from the three-loop level, the wave function renormalization constant depends on the gauge parameter, ξ. The parameter in the above equations is deﬁned through the gluon propagator as i k k Dab (k)= − g − ξ µ ν δab , (36) µν k2 µν k2 where a and b are colour indices. OS OS We want to mention that our results for Zm and Z2 agree with the literature [10, 11]. Whereas in [10,11] they are expressed in terms of the bare coupling we decided to use the renormalized αs as an expansion parameter which to our opinion is more convenient in practical applications. The genuine three-loop integrals which appear in the DREG calculation are the same for DRED. The main complication is the more involved renormalization which is discussed in more detail in the next Section.

3 Renormalization in DRED

Let us in this Section collect the DRED counterterms needed for our calculation. In addition 1 to the strong coupling αs also the evanescent coupling αe has to be renormalized to two- 2 loop order. The evanescent couplings η1, η2 and η3 appear for the ﬁrst time at three-loop order and thus no renormalization is necessary. Both for the heavy quark mass, mq, and the ε-scalar mass, mε, two-loop counterterms are necessary. Whereas the couplings are renormalized using minimal subtraction the masses are renormalized on-shell. The corresponding counterterms are deﬁned through

2 0,DR 2ǫ DR DR 0 2ǫ 2 αs = µ Zs αs , αe = µ (Ze) αe , 0,DR OS,DR 0 2 2 OS mq = MqZm , mε = mεZmε . (37)

We attach an additional index “DR” to the quark mass renormalization constant in order to remind that it relates the pole mass to the bare mass in the DRED scheme3 (in this context see also Ref. [30]). DR Recently the quantities Ze and Zs have been computed to three- and four-loop order [4, 30], respectively. The results have been presented in terms of the corresponding β functions. For completeness we present in the following the two-loop results for the

1We refer to [4, 29] for a precise deﬁnition of the evanescent couplings. 2 For the SU(3) gauge group, which we exclusively consider in this paper, there are three independent such couplings [29]. 3In principle such an index would also be necessary in Section 2. However, since the MS scheme in connection with DREG constitutes the standard framework we refrain from introducing an additional index there.

12 renormalization constants

2 αDR 1 11 1 αDR 1 121 11 ZDR = 1+ s − C + T n + s C2 − C T n s π ǫ 24 A 6 F f π ǫ2 384 A 48 A F f ! 1 1 17 5 1 + n2 T 2 + − C2 + C T n + C T n , (38) 24 f F ǫ 96 A 48 A F f 16 F F f αDR 1 3 α 1 1 1 1 Z = 1+ s − C + e − C + C + T n e π ǫ 4 F π ǫ 4 A 2 F 4 F f 2 αDR 1 11 9 1 1 7 55 + s C C + C2 − C T n + C2 − C C π ǫ2 32 A F 32 F 8 F F f ǫ 256 A 192 A F ! 3 1 5 αDR α 1 3 3 − C2 − C T n + C T n + s e C C − C2 64 F 32 A F f 48 F F f π π ǫ2 8 A F 4 F 3 1 3 5 11 5 − C T n + C2 − C C + C2 + C T n 8 F F f ǫ 32 A 8 A F 16 F 32 F F f α 2 1 3 3 3 3 3 3 + e C2 − C C + C2 − C T n + C T n + T 2 n2 π ǫ2 32 A 8 A F 8 F 16 A F f 8 F F f 32 F f 1 3 5 1 3 3 + − C2 + C C − C2 + C T n − C T n ǫ 32 A 16 A F 4 F 32 A F f 16 F F f α 1 η 9 η 5 η 3 η 2 1 27 η 2 1 15 η η 1 9 + e 1 − 2 − 3 − 1 + 2 + 1 3 π ǫ π 32 π 16 π 16 π ǫ 256 π ǫ 16 π π ǫ 64 η 2 1 21 − 3 . (39) π ǫ 128 Note that we set Nc = 3 in all terms containing the couplings ηi since our implementation of these couplings is only valid for SU(3). We would also like to point out that the DR analytical form of Zs is identical to the corresponding result in the MS scheme. This has been shown by an explicit calculation in Ref. [3]. The one-loop result for Ze can be found in Ref. [29]. OS,DR The one-loop corrections to Zm can be found in Ref. [12] and the two-loop terms have been computed in Ref. [13]. For our calculation we also need the O(ǫ2) and O(ǫ) parts of the one- and two-loop terms, respectively. In Section 4 we want to present the ﬁnite result obtained by considering the ratio OS,DR DR DR Zm /Zm . The quantity Zm has been computed in Ref. [30] to three and in Ref. [4] even to four-loop order. Whereas in [4, 30] only the anomalous dimensions are given we

13 want to present the explicit result for the renormalization constant

2 αDR 1 3 αDR 1 11 9 1 ZDR = 1+ s − C + s C C + C2 − C T n m π ǫ 4 F π ǫ2 32 A F 32 F 8 F F f ! 1 91 3 5 αDR α 3 1 + − C C − C2 + C T n + s e C2 ǫ 192 A F 64 F 48 F F f π π 16 ǫ F 3 2 DR αe 1 1 1 2 1 αs 1 121 2 + CACF − CF − CF TF nf + 3 − CACF π ǫ 16 8 16 π ! ǫ 576 33 9 11 3 1 − C C2 − C3 + C C T n + C2 T n − C T 2 n2 128 A F 128 F 72 A F F f 32 F F f 36 F F f 1 1613 295 9 59 29 + C2 C + C C2 + C3 − C C T n − C2 T n ǫ2 3456 A F 768 A F 256 F 216 A F F f 192 F F f 5 1 10255 133 43 281 + C T 2 n2 + − C2 C + C C2 − C3 + 216 F F f ǫ 20736 A F 768 A F 128 F 2592 1 23 1 35 + ζ C C T n + − ζ C2 T n + C T 2 n2 4 3 A F F f 96 4 3 F F f 1296 F F f 2 αDR α 1 11 15 1 1 5 + s e − C C2 − C3 + C2 T n + C2 C π π ǫ2 192 A F 64 F 48 F F f ǫ 256 A F ! 7 9 3 αDR α 2 1 9 9 + C C2 + C3 − C2 T n + s e − C C2 + C3 32 A F 64 F 32 F F f π π ǫ2 64 A F 32 F 9 1 1 7 3 1 + C2 T n + − C2 C + C C2 − C3 − C C T n 64 F F f ǫ 64 A F 32 A F 8 F 64 A F F f 1 α 3 1 1 1 1 − C2 T n + e − C2 C + C C2 − C3 8 F F f π ǫ2 48 A F 12 A F 12 F 1 1 1 1 1 1 1 + C C T n − C2 T n − C T 2 n2 + C2 C − C C2 + C3 24 A F F f 12 F F f 48 F F f ǫ 32 A F 8 A F 8 F 1 5 1 1 α 2 η 1 − C C T n + C2 T n + C T 2 n2 − e 1 24 A F F f 48 F F f 96 F F f 8 π π ǫ 5 α 2 η 1 1 α 2 η 1 3 α η 2 1 5 α η 2 1 + e 2 + e 3 + e 1 − e 2 36 π π ǫ 12 π π ǫ 64 π π ǫ 12 π π ǫ 7 α η 2 1 1 α η η 1 + e 3 − e 1 3 . (40) 96 π π ǫ 16 π π π ǫ In order to achieve the ﬁnite result for the relation between the pole and the DR quark mass it is necessary to ﬁx a renormalization scheme also for the mass of the ε scalar, mε. Although, there is in general no tree-level term in the Lagrangian, there are loop induced contributions to mε which require the introduction of corresponding counterterms. The relevant Feynman diagrams contributing to the ε-scalar propagator show quadratic

14 (a) (b) (c)

(d) (e) (f)

Figure 3: One- and two-loop Feynman diagrams contributing to the ε-scalar propagator. Dashed lines denote ε scalars, curly lines denote gluons and solid lines denote massive quarks with mass mq. divergences and therefore, one needs to consider only contributions from massive particles. Thus, in our case, only diagrams involving a massive quark have to be taken into account. Some sample diagrams are shown in Fig. 3.

It is common practice to renormalize mε on-shell and require that the renormalized mass is zero to each order in perturbation theory [31]. This scheme is known as the DR′ scheme [31] and offers the advantage that the ε-scalar mass completely decouples from the physical observables. For supersymmetric theories the DR and DR′ renormalization schemes are the same, while for theories with broken supersymmetry the latter one is most convenient. At one-loop order there is only one relevant diagram (cf. Fig. 3(a)) which has to be evaluated for vanishing external momentum. A closer look to the two-loop diagrams shows that they develop infra-red divergences in the limit mε → 0 (cf., e.g., Fig. 3(e)). They can be regulated by introducing a small but non-vanishing mass for the ε scalars. After the 2 2 2 subsequent application of an asymptotic expansion [32] in the limit q = mε ≪ Mq the infra-red divergences manifest themselves as ln(mε) terms. Furthermore, one-loop diagrams like the ones in Fig. 3(b) and (c) do not vanish anymore and have to be taken 2 into account as well. Although they are proportional to mε, after renormalization they 2 induce two-loop contributions which are proportional to Mq , partly multiplied by ln(mε) terms. It is interesting to note that in the sum of the genuine two-loop diagrams and the counterterm contributions the limit mε → 0 can be taken which demonstrates the infra-red finiteness of the on-shell mass of the ε scalar. 2 2 Taking the infra-red finiteness for granted, it is also possible to choose q = mε = 0 from the very beginning. Then the individual diagrams are infra-red divergent, however, the

15 sum is not. We have performed the calculation both ways and checked that the ﬁnal result is the same. It is given by

M 2 α 2 1 q ZOS = 1 − e n T +2+2L + ǫ 2+ π2 +2L + L2 m2 mε π h F ǫ µ 6 µ µ ε 2 αDR 3 1 1 3 αDR α 1 3 3 − s n T + + L C + s e n T C + C π h F 4 ǫ 4 2 µ A π π h F ǫ2 8 A 2 F ! 1 7 3 3 3 15 1 + C + C + C + C L + + π2 C ǫ 8 A 2 F 4 A 2 F µ 8 16 A 3 1 7 3 3 3 + + π2 C + C + C L + C + C L2 2 8 F 4 A 2 F µ 4 A 4 F µ α 2 1 1 1 1 1 1 + e n T C − C − T n + C π h F ǫ2 4 A 2 F 2 F f ǫ 2 F 1 1 5 1 1 − (1 + L ) T n − C + C − + π2 T n 2 µ F f 2 A 2 F 2 24 F f 1 1 1 1 1 − C − 2C + T n L − C − C + T n L2 2 A F 2 F f µ 4 A 2 F 4 F f µ α η 3 1 1 3 3 3 1 3 3 + e 1 n + + L + + π2 + L + L2 π π h 16 ǫ2 ǫ 16 8 µ 16 32 8 µ 8 µ α η 5 1 5 1 25 5 5 − e 2 n + 5+ L + + π2 + 10L + L2 π π h 4 ǫ2 2 µ ǫ 2 24 µ 2 µ α η 7 1 7 7 1 7 7 7 7 − e 3 n + + L + + π2 + L + L2 , (41) π π h 16 ǫ2 16 8 µ ǫ 16 96 8 µ 8 µ where the constants are deﬁned after Eq. (14). The overall factor nh in front of the one- and two-loop corrections shows that the renormalization of mε only inﬂuences those terms which contain a closed heavy quark loop.

OS 4 Zm in dimensional reduction

The approach to extract the on-shell mass counterterm in DRED can be taken over from DREG as described in Section 2, i.e. one considers the inverse quark propagator and requires that it has a zero at the position of the pole. Again, the counterterm diagrams are generated order-by-order in a generic way. The major complication as compared to the calculation in DREG is the appearance of the evanescent couplings and the ε scalars. In particular, there are three diﬀerent four- ε vertices. This leads to many more Feynman diagrams which have to be considered. Whereas in the case of DREG about 130 diagrams contribute there are more than 1100 in

16 (a)(b)(c)

(d)(e)(f)

Figure 4: Sample three-loop diagrams contributing to the quark propagator which have to be considered additionally in case dimensional reduction is used for the regularization. Solid lines denote massive quarks with mass mq, curly lines denote gluons and the ε scalars are represented by dashed lines. In the closed fermion loops all quark ﬂavours have to be considered.

the case of DRED. Typical Feynman diagrams are shown in Fig. 1 and Fig. 4. The more involved renormalization has already been discussed in Section 3. OS,DR We avoid to present the result for the divergent quantity Zm but show the result for the ratio to the DR quantity which is ﬁnite. We cast our result in the following form

ZOS,DR mDR zOS,DR = m = q = 1+ δ(1)zOS,DR + δ(2)zOS,DR + δ(3)zOS,DR . (42) m DR OS m m m Zm Mq

Since our implementation of the couplings ηi is only valid for SU(3) we furthermore set Nc = 3. Our one-, two- and three-loop results read:

αDR 4 1 α δ(1)zOS,DR = − s + L − e , (43) m π 3 µ 3 π 2 αDR 3143 2 1 1 151 7 δ(2)zOS,DR = s − − π2 − π2 ln2+ ζ − L − L2 m π 288 9 9 6 3 24 µ 8 µ ! 71 1 13 1 143 1 13 + + π2 + L + L2 n + − π2 + L 144 18 36 µ 12 µ l 144 9 36 µ 1 αDR α 3 1 α 2 1 1 + L2 n + s e − + L + e + n , (44) 12 µ h π π 8 3 µ π 6 48 f

17 3 αDR 1160387 24707 38 7 67 δ(3)zOS,DR = s − − π2 − π2 ln2+ π2 ln2 2+ ζ m π 10368 2592 9 27 72 3 ! 341 1331 1705 19 76 20089 + π4 + π2ζ − ζ + ln4 2+ a − + π2 2592 432 3 216 5 54 9 4 288 1 3 1475 21 42235 923 11 + π2 ln 2 − ζ L − L2 − L3 + + π2 + π2 ln 2 2 4 3 µ 96 µ 16 µ 3888 648 81 2 707 61 1 8 3463 35 − π2 ln2 2+ ζ − π4 − ln4 2 − a + + π2 81 216 3 1944 81 27 4 432 108 1 7 35 2 2353 13 7 + π2 ln2+ ζ L + L2 + L3 n − + π2 + ζ 27 9 3 µ 16 µ 9 µ l 23328 324 54 3 89 1 13 1 5917 13 2 + + π2 L + L2 + L3 n2 − − π2 − ζ 648 54 µ 216 µ 108 µ l 11664 324 27 3 143 1 13 1 77065 13375 + − π2 L + L2 + L3 n n + − π2 324 54 µ 108 µ 54 µ l h 3888 1944 640 1 751 41 1 5 1 + π2 ln2+ π2 ln2 2 − ζ − π4 + π2ζ − ζ − ln4 2 81 81 216 3 972 4 3 4 5 81 8 4435 23 1 7 35 2 − a + − π2 + π2 ln2+ ζ L + L2 + L3 n 27 4 432 54 27 9 3 µ 16 µ 9 µ h 9481 4 11 197 1 13 1 − − π2 − ζ + − π2 L + L2 + L3 n2 23328 405 54 3 648 27 µ 216 µ 108 µ h 2 αDR α 41105 2 1 7 35 7 + s e + π2 + π2 ln2+ ζ + L + L2 π π 20736 27 27 24 3 12 µ 24 µ ! 27 1 11 1 113 1 11 1 − + π2 + L + L2 n − − π2 + L + L2 n 64 54 54 µ 36 µ l 192 27 54 µ 36 µ h αDR α 2 1397 5 1 55 5 1 + s e − ζ − L + + ζ − L n π π 2592 36 3 6 µ 1728 36 3 48 µ f α 3 7 5 31 5 5 α 2 η + e − − ζ − n + n2 − e 2 π 144 216 3 576 f 576 f 24 π π 9 α η 2 15 α η 2 7 α η 2 3 α η η − e 1 + e 2 − e 3 + e 1 3 . (45) 256 π π 16 π π 128 π π 64 π π π For αe = αs the two-loop result agrees with [13], while the three-loop one is new. There is a very strong cross check of the results in Eqs. (43), (44) and (45) for the limit nh = 0. The starting point is the relation between the MS and the on-shell mass in DREG which can easily be obtained from the results of Section 2 and the MS counterpart of MS MS Eq. (40) (see, e.g., Ref. [33]). In this relation, which depends on αs , both m and MS αs are replaced by their DRED counterparts using Eqs. (4.2) and (4.3) of Ref. [4]. In this DR,OS way we could verify the results for zm |nh=0. This provides a strong consistency check both on the results presented in this paper but also on the approach used in [4] for the

18 extraction of the conversion formulae between the MS and DR quantities. In order to get an impression of the numerical size of the corrections, both in DREG and DRED, let us consider the relation between the minimally subtracted and the pole mass (5),MS for the case of the bottom and top quark. As input we use αs (MZ )=0.1189 [34], Mb =4.800 GeV and Mt = 171.4 GeV [35]. (5),MS (5),MS In the framework of DREG it is straightforward to convert αs (MZ ) to αs (Mb) and (6),MS 4 αs (Mt) using four-loop accuracy leading to

MS bottom : mb (Mb)=3.953 GeV = 4.800(1 − 0.0929 − 0.0493 − 0.0342) GeV = (4.800 − 0.445 − 0.236 − 0.164) GeV , (46) MS top : mt (Mt) = 161.1 GeV = 171.4(1 − 0.0461 − 0.0109 − 0.0033) GeV = (171.4 − 7.9 − 1.9 − 0.6) GeV , (47)

(5),MS (6),MS with αs (Mb)=0.2188 and αs (Mt)=0.1085. The numbers given in the round brackets of the above equations indicate the contributions from the tree-level, one-, two- and three-loop conversion relation. Within DRED the numerical analysis gets more involved since four more couplings appear whose values are needed for µ = Mb and µ = Mt. As mentioned in the Introduction, DRED is an appropriate scheme for supersymmetric theories where in the strong sector only one coupling constant is present — like in usual QCD using DREG. Since all supersymmetric particles are heavier than the electroweak scale it is necessary to match the full theory to the Standard Model at some properly chosen scale µdec. At this step the additional couplings appear. (5) (5) dec For the computation of αe (Mb) and ηi (Mb) we use µ = MZ , evaluate in a ﬁrst step (5),DR αs (MZ ) using the three-loop relation given in Ref. [4] and require

(5),DR (5) (5) αs (MZ )= αe (MZ )= η1 (MZ ) , (5) (5) η2 (MZ )= η3 (MZ)=0 . (48)

DR In a second step the renormalization group functions, which are known to four (αs ), three (αe) and one-loop order (ηi) [4,30] are used to obtain

(5),DR αs (Mb) = 0.2241 , (5) αe (Mb) = 0.1721 , (5) η1 (Mb) = 0.2152 , (5) η2 (Mb) = −0.01798 , (5) η3 (Mb) = −0.005777 . (49)

4 We use RunDec [36] for the running and decoupling of αs in the MS scheme.

19 dec In the case of the top quark we choose µ = Mt. It is necessary to know the couplings (6),MS for six active ﬂavours and thus we evaluate in a ﬁrst step αs (MZ )=0.1178 and pose the analogue requirements as in Eq. (48) with “(5)” replaced by “(6)” and MZ by Mt. This leads to

(6),DR (6) (6) αs (Mt)= αe (Mt)= η1 (Mt)=0.1096 , (6) (6) η2 (Mt)= η3 (Mt)=0 . (50)

Finally, we can insert the results of Eqs. (49) and (50) into Eq. (42) and obtain

DR bottom : mb (Mb)=3.859 GeV = 4.800(1 − 0.1134 − 0.0495 − 0.033) GeV = (4.800 − 0.544 − 0.238 − 0.159) GeV , (51) DR top : mt (Mt) = 159.0 GeV = 171.4(1 − 0.0581 − 0.0105 − 0.0030) GeV = (171.4 − 10.0 − 1.8 − 0.5) GeV . (52)

The perturbative expansion shows a similar behaviour as for the MS–on-shell mass relation: the three-loop terms amount to 159 MeV and 500 MeV, respectively, and are thus far above the current uncertainty for the bottom quark mass [37] and much larger than the expected uncertainty for the top quark mass [38].

5 Conclusions

The main result of this paper is the relation between the pole and DR quark mass to three-loop order in QCD where dimensional reduction has been used as a regularization scheme. The conversion formula has been obtained in analytical form where all occurring integrals have been reduced to a small set of master integrals with the help of the Laporta algorithm. Due to the occurrence of evanescent couplings when using DRED within QCD it is more advantageous to use in this case dimensional regularization. However, the latter cannot be used in supersymmetric models. Thus, our result constitutes an important preparation for similar calculations in supersymmetric extensions of the Standard Model. As a by-product we have obtained the corresponding relation and the on-shell wave function renormalization using dimensional regularization. This constitutes the first check of the analytical results obtained about seven years ago. Furthermore, we can confirm the OS observation of Ref. [11] that Z2 depends on the gauge fixing parameter starting from three-loop order.

Acknowledgements We would like to thank Stefan Bekavac for discussions about the Mellin-Barnes method and checking some of our results. This work was supported by the “Impuls- und Vernet- zungsfonds” of the Helmholtz Association, contract number VH-NG-008 and the DFG through SFB/TR 9. The Feynman diagrams were drawn with JaxoDraw [39].

20 References

[1] W. A. Bardeen, A. J. Buras, D. W. Duke and T. Muta, Phys. Rev.D 18 (1978) 3998.

[2] W. Siegel, Phys. Lett. B 84 (1979) 197.

[3] D. M. Capper, D. R. T. Jones and P. van Nieuwenhuizen, Nucl. Phys. B 167 (1980) 479.

[4] R. V. Harlander, D. R. T. Jones, P. Kant, L. Mihaila and M. Steinhauser, JHEP 0612, 024 (2006) [arXiv:hep-ph/0610206].

[5] R. Tarrach, Nucl. Phys. B 183 (1981) 384.

[6] N. Gray, D. J. Broadhurst, W. Grafe and K. Schilcher, Z. Phys. C 48 (1990) 673.

[7] D. J. Broadhurst, N. Gray and K. Schilcher, Z. Phys. C 52 (1991) 111.

[8] K. G. Chetyrkin and M. Steinhauser, Phys. Rev. Lett. 83 (1999) 4001 [arXiv:hep- ph/9907509].

[9] K. G. Chetyrkin and M. Steinhauser, Nucl. Phys. B 573 (2000) 617 [arXiv:hep- ph/9911434].

[10] K. Melnikov and T. van Ritbergen, Phys. Lett. B 482 (2000) 99 [arXiv:hep- ph/9912391].

[11] K. Melnikov and T. van Ritbergen, Nucl. Phys. B 591 (2000) 515 [arXiv:hep- ph/0005131].

[12] S. P. Martin and M. T. Vaughn, Phys. Lett. B 318 (1993) 331 [arXiv:hep- ph/9308222].

[13] L. V. Avdeev and M. Y. Kalmykov, Nucl. Phys. B 502 (1997) 419 [arXiv:hep- ph/9701308].

[14] P. Marquard, J. H. Piclum, D. Seidel and M. Steinhauser, Nucl. Phys. B 758 (2006) 144 [arXiv:hep-ph/0607168].

[15] J. H. Piclum, Dissertation, Universit¨at Hamburg, in preparation.

[16] P. Nogueira, J. Comput. Phys. 105 (1993) 279.

[17] R. Harlander, T. Seidensticker and M. Steinhauser, Phys. Lett. B 426 (1998) 125 [hep-ph/9712228].

[18] T. Seidensticker, hep-ph/9905298.

21 [19] S. Laporta and E. Remiddi, Phys. Lett. B 379 (1996) 283 [arXiv:hep-ph/9602417].

[20] S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087 [arXiv:hep-ph/0102033].

[21] P. Marquard and D. Seidel, unpublished.

[22] C. Bauer, A. Frink and R. Kreckel, arXiv:cs.sc/0004015.

[23] R. H. Lewis, Fermat’s User Guide, http://www.bway.net/˜lewis.

[24] M. Tentyukov and J. A. M. Vermaseren, arXiv:cs.sc/0604052.

[25] K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys. B 192 (1981) 159.

[26] V. A. Smirnov, Phys. Lett. B 460 (1999) 397 [arXiv:hep-ph/9905323].

[27] J. B. Tausk, Phys. Lett. B 469 (1999) 225 [arXiv:hep-ph/9909506].

[28] M. Czakon, Comput. Phys. Commun. 175 (2006) 559 [arXiv:hep-ph/0511200].

[29] I. Jack, D. R. T. Jones and K. L. Roberts, Z. Phys. C 62 (1994) 161 [arXiv:hep- ph/9310301].

[30] R. Harlander, P. Kant, L. Mihaila and M. Steinhauser, JHEP 0609, 053 (2006) [arXiv:hep-ph/0607240].

[31] I. Jack, D. R. T. Jones, S. P. Martin, M. T. Vaughn and Y. Yamada, Phys. Rev. D 50 (1994) 5481 [arXiv:hep-ph/9407291].

[32] V. A. Smirnov, “Applied asymptotic expansions in momenta and masses,” Springer Tracts Mod. Phys. 177 (2002) 1.

[33] K. G. Chetyrkin, Nucl. Phys. B 710 (2005) 499 [arXiv:hep-ph/0405193].

[34] S. Bethke, arXiv:hep-ex/0606035.

[35] Tevatron Electroweak Working Group, hep-ex/0608032.

[36] K. G. Chetyrkin, J. H. K¨uhn and M. Steinhauser, Comput. Phys. Commun. 133 (2000) 43 [arXiv:hep-ph/0004189].

[37] J. H. K¨uhn, M. Steinhauser and C. Sturm, arXiv:hep-ph/0702103.

[38] M. Martinez and R. Miquel, Eur. Phys. J. C 27 (2003) 49 [arXiv:hep-ph/0207315].

[39] D. Binosi and L. Theussl, Comput. Phys. Commun. 161 (2004) 76 [arXiv:hep- ph/0309015].