Renormalization Constants and Beta Functions for the Gauge Couplings of the Standard Model to Three-Loop Order

Home , Beta function (physics), Coupling constant

SFB/CPP-12-61, TTP12-30

Renormalization constants and beta functions for the gauge couplings of the Standard Model to three-loop order

Luminita N. Mihaila, Jens Salomon, Matthias Steinhauser1 1Institut f¨ur Theoretische Teilchenphysik, Karlsruhe Institute of Technology (KIT), D-76128 Karlsruhe, Germany We compute the beta functions for the three gauge couplings of the Standard Model in the minimal subtraction scheme to three loops. We take into account contributions from all sectors of the Standard Model. The calculation is performed using both Lorenz gauge in the unbroken phase of the Standard Model and background ﬁeld gauge in the spontaneously broken phase. Furthermore, we describe in detail the treatment of γ5 and present the automated setup which we use for the calculation of the Feynman diagrams. It starts with the generation of the Feynman rules and leads to the bare result for the Green’s function of a given process.

PACS numbers: 11.10.Hi 11.15.Bt 2

I. INTRODUCTION

Renormalization group functions are fundamental quantities of each quantum ﬁeld theory. They provide insights in the energy dependence of cross sections, hints to phase transitions and can provide evidence to the energy range in which a particular theory is valid. The renormalization group functions of the gauge couplings in the Standard Model (SM) are of particular importance in the context of Grand Uniﬁed Theories allowing the extrapolation of low-energy precision data to high energies, not accessible by collider experiments. Important milestones for the calculation of the gauge coupling beta functions in the Standard Model are the following computations:

The one-loop beta functions in gauge theories along with the discovery of asymptotic • freedom have been presented in Refs. [1, 2].

The corresponding two-loop corrections • – in gauge theories without fermions [3, 4], – in gauge theories with fermions neglecting Yukawa couplings [5–7], – with corrections involving Yukawa couplings [8],

are also available.

The two-loop gauge coupling beta functions in an arbitrary quantum ﬁeld theory have • been considered in Ref. [9, 10].

The contribution of the scalar self-interaction at three-loop order has been computed • in [11, 12].

The gauge coupling beta function in quantum chromodynamics (QCD) to three loops • is known from Ref. [13, 14].

The three-loop corrections to the gauge coupling beta function involving two strong • and one top quark Yukawa coupling have been computed in Ref. [15].

The three-loop corrections for a general quantum ﬁeld theory based on a single gauge • group have been computed in [16].

The four-loop corrections in QCD are known from Refs. [17, 18]. • Two-loop corrections to the renormalization group functions for the Yukawa and Higgs boson self-couplings in the Standard Model are also known [10, 19–23]. Recently the dominant three-loop corrections to the renormalization group functions of the top quark Yukawa and the Higgs boson self-coupling have been computed in Ref. [24]. In this calculation the gauge couplings and all Yukawa coupling except the one of the top quark are set to zero. In this paper we present details to the three-loop calculation of the gauge coupling renormalization constants and the corresponding beta functions in the SM taking into account all sectors. The results have already been presented in Ref. [25]. The remainder of the paper is organized as follows: In the next Section we introduce our notation and describe in detail how we proceed to obtain the beta functions of the gauge couplings. In particular, we describe the calculation in Lorenz gauge and background ﬁeld 3

gauge (BFG), discuss our setup for an automated calculation, and explain our treatment of γ5. The analytical results for the beta functions are presented in Section III. In contrast to Ref. [25] we show the results including all Yukawa couplings. Section IV is devoted to a description of the checks which have been performed to verify our result. A discussion of the numerical impact of the newly obtained corrections is given in Section V. We conclude in Section VI. Explicit results for the renormalization constants are relegated to Appendix A and Appendix B. In Appendix C we present three-loop results for the beta functions of the QED coupling constant and the weak mixing angle. Furthermore, we present in Appendix D translation rules which are useful in order to compare parts of our ﬁndings with the results of Ref. [16].

II. THE CALCULATION OF THE BETA FUNCTIONS

In this paper we present the beta functions for the three gauge couplings of the SM up to three loops in the modiﬁed minimal subtraction (MS) renormalization scheme. In the corresponding calculation we took into account contributions involving the three gauge couplings of the SM, the top, the bottom, and the tau Yukawa couplings and the Higgs self-coupling. Let us mention that we were able to derive the beta functions for a general SM Yukawa sector from the calculation involving the aforementioned seven couplings. We postpone the discussion of this issue to the next Section. In the following, we give details on the computation at the three-loop order. We deﬁne the beta functions as d α µ2 i = β ( α , ǫ) , (1) dµ2 π i { j}

where ǫ = (4 d)/2 is the regulator of Dimensional Regularization with d being the space- time dimension− used for the evaluation of the momentum integrals. The dependence of the couplings αi on the renormalization scale is suppressed in the above equation. The three gauge couplings α1, α2 and α3 used in this paper are related to the quantities usually used in the SM by

5 αQED α1 = 2 , 3 cos θW αQED α2 = 2 , sin θW α3 = αs , (2)

where αQED is the ﬁne structure constant, θW the weak mixing angle and αs the strong coupling. We adopt the SU(5) normalization which leads to the factor 5/3 in the equation for α1. The Yukawa couplings are deﬁned as

2 αQEDmx αx = 2 2 with x = t, b, τ , (3) 2 sin θW MW where mx and MW are the fermion and W boson mass, respectively. 4

We denote the Higgs boson self-coupling by λˆ, where the Lagrange density contains the following term = . . . (4πλˆ)(H†H)2 + ... , (4) LSM − describing the quartic Higgs boson self-interaction. The beta functions are obtained by calculating the renormalization constants relating bare and renormalized couplings via αbare = µ2ǫZ ( α , ǫ)α . (5) i αi { j} i bare Taking into account that αi does not depend on µ, Eqs. (1) and (5) lead to

7 −1 αi αi ∂Zαi αi ∂Zαi βi = ǫ + βj 1+ , (6) − π Zα ∂αj Zα ∂αi " i j ,j6 i # i =1X= ˆ where i =1, 2 or 3. We furthermore set α4 = αt, α5 = αb, α6 = ατ and α7 = λ. The first term in the first factor of Eq. (6) originates from the term µ2ǫ in Eq. (5) and vanishes in four space-time dimensions. The second term in the first factor contains the beta functions of the remaining six couplings of the SM. Note that (for the gauge couplings) the

one-loop term of Zαi only contains αi, whereas at two loops all couplings are present except λˆ. The latter appears for the first time at three-loop level. As a consequence, it is necessary to know βj for j = 4, 5, 6 to one-loop order and only the ǫ-dependent term for β7, namely β = ǫα /π. From the second term in the first factor and the second factor of Eq. (6) one 7 − 7 can read off that three-loop corrections to Zαi are required for the computation of βi to the same loop order. We have followed two distinct paths to obtain the results for the three-loop renormalization constants, which we discuss in the following Subsections, where we mention features and differences.

A. Lorenz gauge in the unbroken phase of the SM

The first method used for the calculation of the renormalization constants is based on Feynman rules derived for the SM in the unbroken phase in a general Lorenz gauge with three independent gauge parameters corresponding to the three simple gauge groups. All building blocks of our calculation are evaluated for general gauge parameters in order to have a strong check of the final results for the beta functions which have to be gauge parameter independent. It is possible to use the unbroken phase of the SM since the beta function in the MS scheme is independent of all mass parameters and thus the spontaneous symmetry breaking does not affect the final result. Note that this choice is advantageous for the calculation because in the unbroken phase much less different types of vertices have to be considered as compared to the phase in which the spontaneous symmetry breaking is present. In principle each vertex containing the coupling gi = √4παi can be used in order to determine the corresponding renormalization constant via

2 (Zvrtx) Zαi = , (7) k Zk,wf Q 5

where Zvrtx stands for the renormalization constant of the vertex and Zk,wf for the wave function renormalization constant; k runs over all external particles.

We have computed Zα3 using both the ghost-gluon and the three-gluon vertex. Zα2 has + − been evaluated with the help of the ghost-W3, the W1W2W3 and the φ φ W3 vertex where φ± is the charged component of the Higgs doublet corresponding to the charged Goldstone boson in the broken phase and W1, W2 and W3 are the components of the W boson. As to

Zα1 , a Ward identity guarantees that there is a cancellation between the vertex and some of the wave function renormalization constants yielding 1 Zα1 = , (8) ZB

where ZB is the wave function renormalization constant for the gauge boson of the U(1) subgroup of the SM in the unbroken phase. In Fig. 1 we show several one-, two- and three-loop sample diagrams contributing to the considered two- and three-point functions. We have not used vertices involving fermions as external particles as they may lead to problems in connection with the treatment of γ in d = 4 dimensions. The vertices selected 5 6 by us are safe in this respect. A detailed discussion of our prescription for γ5 is given below. In order to compute the individual renormalization constants entering Eq. (7) we proceed as outlined, e.g., in Ref. [15]. The underlying formula can be written in the form

Z =1 K (Z Γ) , (9) Γ − ǫ Γ where Γ represents the two- or three-point function corresponding to the renormalization constant ZΓ and the operator Kǫ extracts the pole part of its argument. From the structure of Eq. (9) it is clear that ZΓ is computed order-by-order in perturbation theory in a recursive way. It is understood that the bare parameters entering Γ on the right-hand side are replaced by the renormalized ones before applying Kǫ. The corresponding counterterms are only needed to lower loop orders than the one which is requested for Γ. In our approach the three-

loop calculation of Zαi requires — besides the result for Zαi to two loops — the one-loop renormalization constants for the other two gauge and the Yukawa couplings. Furthermore we have to renormalize the gauge parameters; the corresponding renormalization constants are given by the wave function renormalization constants of the corresponding gauge bosons which we anyway have to evaluate in the course of our calculation.

B. Background ﬁeld gauge in the spontaneously broken phase

The second method that we used in order to get an independent result for the renormalization constants of the gauge couplings is a calculation in the BFG [26, 27]. The basic idea of the BFG is the splitting of all gauge ﬁelds in a “quantum” and a “classical” part where in practical calculations the latter only occurs as external particle. The BFG has the advantage that Ward identities guarantee that renormalization constants for gauge couplings can be obtained from the exclusive knowledge of the corresponding wave function renormalization constant.1 Thus we have the following formula

1 In Lorenz gauge, this only works for U(1) gauge groups, cf. Subsection II A. 6

FIG. 1. Sample Feynman diagrams contributing to the Green’s functions which have been used for the calculation of the renormalization constants of the gauge couplings. Solid, dashed, dotted, curly and wavy lines correspond to fermions, Higgs bosons, ghosts, gluons and electroweak gauge bosons, respectively. 7

1 Zαi = , (10) ZAi,wf

where A denotes the gauge boson corresponding to the gauge coupling αi. In contrast to the calculation using Lorenz gauge, we performed the calculation in the BFG in the spontaneously broken phase of the SM. As discussed in the last Subsection, such a calculation is more involved than a calculation in the unbroken phase since more vertices are present. On the other hand it constitutes an additional check of our result, that allows us not only to compare the BFG and the Lorenz gauge but also to switch from the broken to the unbroken phase of the SM. Since the calculation has been performed in the broken phase we have computed the transverse part of the two-point functions of the (background) photon, Z boson, photon-Z mixing, W boson and gluon which we denote by Πγ , ΠZ, ΠγZ , ΠW , Πg, respectively. Sample Feynman diagrams up to three loops can be found in the ﬁrst two lines of Fig. 1. ΠW and Πg can be used in analogy to Subsection II A in order to obtain the corresponding renormalization constants which leads in combination with Eq. (10) to the renormalization constants for α2 and αs. We found complete agreement with the calculation performed in Lorenz gauge. As far as the self energies involving photon and Z boson are concerned, we consider at the bare level appropriate linear combinations in order to obtain the B and W boson self-energy contributions. To be precise, we have

2 bare bare bare 2 bare ΠB = cos θW Πγ + 2 cos θW sin θW ΠγZ + sin θW ΠZ , Π = sin2 θbareΠ 2 cos θbare sin θbareΠ + cos2 θbareΠ . (11) W W γ − W W γZ W Z The second linear combination can immediately be compared with the bare result obtained from the charged W boson self energy and complete agreement up to the three-loop order has been found. This constitutes a strong consistency check on the implementation of the BFG Feynman rules. ΠB is used together with Eq. (10) in order to obtain the renormalization constant for α1. Again, complete agreement with the calculation described in the previous Subsection has been found. In our BFG calculation we want to adopt Landau gauge in order to avoid the renormalization of the gauge parameters ξi. However, it is not possible to choose Landau gauge from the very beginning since some Feynman rules for vertices involving a background gauge boson contain terms proportional to 1/ξi where ξi = 0 corresponds to Landau gauge. To circumvent this problem we evaluate the bare integrals for arbitrary gauge parameters. In the ﬁnal result all inverse powers of ξi cancel and thus the limit ξi = 0 can be taken at the bare level.

C. Automated Calculation

Higher order calculations in the SM taking into account all contributions are quite involved. Apart from the complicated loop integrals, there are many diﬀerent Feynman rules and plenty of Feynman diagrams which have to be considered. In our calculation we have used a setup which to a large extend avoids manual interventions in order to keep the error-proneness to a minimum. 8

QGRAF q2e exp MATAD/MINCER

FeynArtsToQ2E

FeynArts model ﬁle

FeynRules

FIG. 2. Overview of our automated setup. Calling up the programs in the uppermost line de- termines and evaluates a given process in a given model. The vertical workﬂow leads to the implementation of a new model in the setup. The programs are discussed in more detail in the text.

As far as the loop integrals are concerned we exploit the fact that the beta function in the MS scheme is independent of the external momenta and the particle masses. Thus, we can choose a convenient kinematical configuration which leads to simple loop integrals as long as the infra-red structure is not modified. In our case we set all particle masses to zero and only keep one external momentum different from zero. We have checked that no infra-red divergences are introduced as we will discuss in detail in Section IV. In this way all loop-integrals are mapped to massless two-point functions that up to three loops can be computed with the help of the package MINCER [28]. As core of our setup we use a well-tested chain of programs that work hand-in- hand: QGRAF [29] generates all contributing Feynman diagrams. The output is passed via q2e [30, 31], which transforms Feynman diagrams into Feynman amplitudes, to exp [30, 31] that generates FORM [32] code. The latter is processed by MINCER [28] and/or MATAD [33] that compute the Feynman integrals and output the ǫ expansion of the result. The paral- lelization of the latter part is straightforward as the evaluation of each Feynman diagram corresponds to an independent calculation. We have also parallelized the part performed by q2e and exp which is essential for our calculation since it may happen that a few times 105 diagrams contribute at three-loop level to a single Green’s function. The described workflow is illustrated on the top of Fig. 2. In order to perform the calculation described in this paper we have extended the above setup by the vertical program chain in Fig. 2. The core of the new part is the program FeynArtsToQ2E which translates FeynArts [34] model files into model files processable by QGRAF and q2e. In this way we can exploit the well-tested input files of FeynArts in our effective and flexible setup based on QGRAF, q2e, exp and MINCER. This avoids the coding of the Feynman rules by hand which for the SM would require a dedicated debugging process. For the part of our calculation based on the BFG we have used the FeynArts model files which come together with version 3.5. However, for Lorenz gauge in the unbroken phase there is no publicly available FeynArts model. For this reason we have used the package FeynRules [35] in order to generate such a file which is also indicated in Fig. 2. Let us mention that FeynArtsToQ2E is not restricted to the SM but can process all model 9

ﬁles available for FeynArts.

D. Treatment of γ5

An important issue in multi-loop calculations is the definition of γ5 away from d = 4 dimensions. A first possibility is the naive regularization that requires that γ5 anti-commutes with all other γ-matrices. This approach has the advantage that its implementation is very simple. However, it can lead to wrong results, especially for Feynman diagrams involving several fermion loops. For example, the naive regularization of γ5 leads to the problematic result (see, e.g., Ref. [36]) tr(γµγνγργσγ )=0 (d = 4) . (12) 5 6 The limit of this expression for d 4 does not agree with its value in the physical case, when the regularization is turned off→ tr(γµγνγργσγ )= 4iεµνρσ (d = 4) . (13) 5 − Here the totally anti-symmetric Levi-Civita tensor is defined by ε0123 = 1. It is therefore reassuring that one can show explicitly that in the computation via the ghost-ghost-gauge boson vertex2 all contributions stemming from this kind of traces vanish. To prove this, we notice that this kind of traces can only lead to non-vanishing contributions if there are at least two of them in a diagram. Only in this case the ε-tensors originating from Eq. (13) can be contracted, providing Lorentz structures that may contribute to the renormalization constants. We observe that the fermion loops can only yield problematic non-vanishing contributions if at least three lines are attached to them. Otherwise, there are too few external momenta and too few open Lorentz indices available. A general three-loop diagram with at least two closed fermion loops has the following form ...... Here the solid lines represent fermions while the double lines can be either scalar bosons or gauge bosons. One can easily show that one cannot attach ghosts as external particles in the above diagram, since there are no vertices involving ghosts and fermions. Thus, the ghost self-energy and the ghost-ghost-gauge boson vertex do not contain diagrams of this type. Therefore, we need to consider only diagrams with two external gauge bosons for the following discussion. So the diagrams which still have to be discussed have the structure

2 We restrict the discussion in this Subsection to the ghost-ghost-gauge boson vertex. For all other vertices without external fermions the reasoning is in complete analogy. 10

There are three ways to replace the double lines:

. The ﬁrst diagram type cannot yield problematic contributions for the same reason as mentioned above. The second diagram type vanishes as the fermion traces involve exactly ﬁve γ-matrices, not counting γ5 matrices. (Remember that we deal with diagrams in which all the propagators are massless.) Finally, the third diagram type involves fermion loops with three external gauge bosons. Such diagrams can indeed contain contributions originating from traces of γ5 and an even number of other γ-matrices. However, the sum over all possible fermion species that can circulate in the loops, including also the diagrams in which the fermions circle in opposite directions, vanishes.3 This is of course a consequence of the cancellation of the Adler-Bell-Jackiw anomaly [38, 39] within the SM, as required by gauge

invariance. Therefore, we are allowed to calculate the Feynman diagrams contributing to Zαi using a naive regularization prescription for γ5, in which the diagrams containing triangle anomalies are set to zero from the very beginning, according to Eq. (12) . As an additional check of the calculation we implemented also a “semi-naive” regulariza- µ ν ρ σ tion prescription for γ5. Explicitly, we evaluate the expression tr(γ γ γ γ γ5) by applying the formal replacement tr(γµγνγργσγ )= 4i˜εµνρσ + (ǫ) . (14) 5 − O The tensorε ˜µνρσ has some similarities with the four-dimensional Levi-Civita tensor: (i) it is completely antisymmetric in all indices; (ii) when contracted with a second one of its kind one obtains the following result

[µ ν ρ σ ] µνρσ ′ ′ ′ ′ ε˜ ε˜µ ν ρ σ = g[µ′ g ν′ g ρ′ g σ′] , (15) where the square brackets denote complete anti-symmetrization. When taking the limit d 4,ε ˜µνρσ converts into the four-dimensional Levi-Civita tensor and Eqs. (14) and (15) ensure→ that it provides the correct four-dimensional result. At this point a comment on Eqs. (14) is in order. It is straightforward to see that the combination of this equation and the cyclic property of traces leads to an ambiguity of order (ǫ). Therefore, we made sure that the terms that need to be treated in this way generate at Omost simple poles in ǫ and the above procedure can be applied directly without introducing additional ﬁnite counterterms.

Let us stress again that we ﬁnd the same result for the renormalization constants Zαi both from the ghost–ghost–gauge boson vertex and by using other vertices and both by applying the “naive” as well as the “semi-naive” scheme. These ﬁndings strongly support the above reasoning.

E. Comparison of the methods

This Subsection is devoted to a brief comparison of the calculation via the Lorenz gauge and the one involving the BFG. As has been mentioned before, in the BFG it is suﬃcient

3 The proof of this statement can be based only on considerations about group theoretic invariants. For details see Chapter 20 of Ref. [37]. 11

Lorenz gauge BFG # loops 1 2 3 4 # loops 1 2 3 4 BB 14 410 45 926 7 111 021 γBγB 13 416 61 968 13 683 693 B B W3W3 17 502 55 063 8 438 172 γ Z 13 604 100 952 23 640 897 gg 9 188 17 611 2 455 714 ZBZB 20 1064 183 465 44 049 196 +B −B cgc¯g 1 12 521 46 390 W W 18 1438 252 162 42 423 978 B B cW3 c¯W3 2 42 2 480 251 200 g g 10 186 17 494 2 775 946 φ+φ− 10 429 46 418 6 918 256 BBB 34 2 172 358 716 73 709 886

W1W2W3 34 2 216 382 767 79 674 008 ggg 21 946 118 086 20 216 024

cgc¯gg 2 66 4 240 460 389

cW1 c¯W2 W3 2 107 10 577 1 517 631 + − φ φ W3 24 2 353 387 338 77 292 771

TABLE I. The number of Feynman diagrams contributing to the Green’s functions evaluated in this work. Left table: two- and three-point functions computed in Lorenz gauge; right table: two-point functions computed in BFG. The superscript “B” denotes background ﬁelds. The ﬁrst column indicates the external legs of the Green’s function, the other columns show the number of diagrams at the individual loop orders. Note that the BBB vertex is computed in order to have a cross check as we will explain in Section IV.

to consider only the gauge boson propagators. This is advantageous as the use of Lorenz gauge also requires the evaluation of three- (or four-) point functions and in most cases it also demands the consideration of additional two-point functions apart from the gauge boson ones. The disadvantages of the BFG are the increased number of vertices and the more involved structure of the vertices containing a background field. In Tab. I we list the number of diagrams for each Green’s function contributing to the one-, two- and three-loop order. The number of diagrams computed in this work is obtained from the sum of the numbers in these columns. For comparison we also provide the corresponding number of diagrams which contribute to the four-loop order. It is tempting to compare the number of contributing Feynman diagrams in Lorenz gauge and in BFG which is, however, not straightforward since we use the former in the broken and the latter in the unbroken phase. Nevertheless, one observes that in the case of β3 the number of diagrams entering the BFG calculation is roughly the same as in case the gluon- ghost vertex is used in Lorenz gauge, even up to four-loop order. All other vertices lead to significantly more diagrams. In the case of β2 there are about three to four times more diagrams to be considered in the BFG as compared to Lorenz gauge. Whereas at three- loop order the difference between approximately 70 000 and 250 000 diagrams is probably not substantial it is striking at four-loop order where the number of Feynman diagrams

goes from about 10000000 (W3W3 , cW3 c¯W3 and cW1 c¯W2 W3 Green’s function) to 42 000 000 (W +BW −B Green’s function) when switching from Lorenz gauge to BFG. Thus, starting from four loops it is probably less attractive to use the BFG. Let us add that the precise number of Feynman diagrams depends on the detailed setup as, e.g., on the implementation of the four-particle vertices. Because of the colour structure 12 we split in our calculation the four-gluon vertex into two cubic vertices by introducing non- propagating auxiliary particles, whereas all other four-particle vertices are left untouched. Let us ﬁnally mention that the CPU time for the evaluation of an individual diagram ranges from less than a second to few minutes. For general gauge parameters it may take up to the order of an hour. Thus the use of about 100 cores leads to a wall-clock time which ranges from a few hours for a calculation in Feynman gauge up to about one day for general gauge parameters. For the preparation of the FORM ﬁles using QGRAF, q2e and exp also a few hours of CPU time are needed which is because of the large amount of Feynman diagrams. The use of about 100 cores leads to a wall-clock time of a few minutes.

III. ANALYTICAL RESULTS

In this Section we present the analytical results for the beta functions. As mentioned before, we are able to present the results involving all contributions of the SM Yukawa sector. The SM Yukawa interactions are described by (see, e.g., Chapter 11 of Ref. [40])

= Q¯LY U ǫH⋆uR Q¯LY DHdR L¯LY LHlR + h.c. , (16) LYukawa − i ij j − i ij j − i ij j where Y U,D,L are complex 3 3 matrices, i, j are generation labels, H denotes the Higgs ﬁeld and ǫ is the 2 2 antisymmetric× tensor. QL, LL are the left-handed quark and lepton doublets, and uR,dR×, lR are the right-handed up- and down-type quark and lepton singlets, respectively. The physical mass-eigenstates are obtained by diagonalizing Y U,D,L by six U,D,L unitary matrices VL,R as follows

˜ f f f f† Ydiag = VL Y VR , f = U, D, L . (17) As a result the charged-current W ± couples to the physical quark states with couplings U D† parametrized by the Cabibbo-Kobayashi-Maskawa (CKM) matrix VCKM VL VL . We furthermore introduce the notation ≡

1 † 1 † 1 † Tˆ = Y U Y U , Bˆ = Y DY D , Lˆ = Y LY L . (18) 4π 4π 4π In order to reconstruct the results for a general Yukawa sector, we multiplied each Feyn- m man diagram by a factor (nh) , where m denotes the number of fermion loops involving Yukawa couplings. After analyzing the structure of the diagrams that can arise, we could establish the following set of replacements that have to be performed in order to take into account a generalized Yukawa sector

n α trTˆ , n α trB,ˆ h t → h b → n α trL,ˆ n α2 trTˆ2, h τ → h t → n α2 trBˆ2, n α α trTˆB,ˆ h b → h t b → n α2 trLˆ2, n2 α2 (trTˆ)2, h τ → h t → n2 α2 (trBˆ)2, n2 α2 (trLˆ)2, h b → h τ → n2 α α trTˆtrB,ˆ n2 α α trTˆtrL,ˆ h t b → h t τ → n2 α α trBˆtrL.ˆ (19) h b τ → 13

Of course, only traces over products of Yukawa matrices can occur because they arise from closed fermion loops. Using Eqs. (17) and (18) it is straightforward to see that in Eq. (19) only traces of diagonal matrices have to be taken except for trTˆBˆ which is given by

αu 0 0 αd 0 0 trTˆBˆ = tr 0 α 0 V 0 α 0 V † . (20)  c  CKM  s  CKM 0 0 αt 0 0 αb      The addition of a fourth generation of fermions to the SM particle content can be also easily accounted for by this general notation. In this case, the Yukawa matrices become 4 4 dimensional. If we assume that the fourth generation is just a repetition of the existing generation× pattern but much heavier and if we neglect all SM Yukawa interactions, then the explicit form of Yukawa matrices reads

0 0 Fˆ = 3×3 , with F = T,B,L. (21) 4 0 α F Here T and B stand for the up- and down-type heavy quarks, and L for the heavy charged leptons, while αF denotes the corresponding Yukawa couplings as deﬁned in Eq. (3). Since in our calculation no Yukawa couplings for neutrinos have been introduced we cannot in- corporate heavy neutrinos. This would require a dedicated calculation which, however, does not pose any principle problem. We are now in the position to present the results for the beta functions of the gauge couplings which are given by

α2 2 16n β = 1 + G 1 2 5 3 (4π) α2 18α 18α 34trTˆ 76α 12α 176α + 1 1 + 2 2trBˆ 6trLˆ + n 1 + 2 + 3 3 25 5 − 5 − − G 15 5 15 (4π) α2 489α2 783α α 3401α2 54α λˆ 18α λˆ 36λˆ2 2827α trTˆ + 1 1 + 1 2 + 2 + 1 + 2 1 4 2000 200 80 25 5 − 5 − 200 (4π) 471α2trTˆ 116α3trTˆ 1267α1trBˆ 1311α2trBˆ 68α3trBˆ 2529α1trLˆ − 8 − 5 − 200 − 40 − 5 − 200 1629α trLˆ 183trBˆ2 51(trBˆ)2 157trBˆtrLˆ 261trLˆ2 99(trLˆ)2 2 + + + + + − 40 20 10 5 20 10 3trTˆBˆ 339trTˆ2 177trTˆtrBˆ 199trTˆtrLˆ 303(trTˆ)2 + + + + + 2 20 5 5 10 232α2 7α α 166α2 548α α 4α α 1100α2 + n 1 1 2 + 2 1 3 2 3 + 3 G − 75 − 25 15 − 225 − 5 9 836α2 44α2 1936α2 + n2 1 2 3 , (22) G − 135 − 15 − 135

2 α2 86 16nG β2 = + π 2 − 3 3 (4 ) 14

α2 6α 518α 4α 196α + 2 1 2 6trTˆ 6trBˆ 2trLˆ + n 1 + 2 + 16α 3 5 − 3 − − − G 5 3 3 (4π) α2 163α2 561α α 667111α2 6α λˆ 593α trTˆ + 2 1 + 1 2 2 + 1 +6α λˆ 12λˆ2 1 4 400 40 − 432 5 2 − − 40 (4π) 729α trTˆ 533α trBˆ 729α trBˆ 51α trLˆ 2 28α trTˆ 1 2 28α trBˆ 1 − 8 − 3 − 40 − 8 − 3 − 8 243α trLˆ 57trBˆ2 45(trBˆ)2 19trLˆ2 5(trLˆ)2 27trTˆBˆ 2 + + + 15trBˆtrLˆ + + + − 8 4 2 4 2 2 57trTˆ2 45(trTˆ)2 + + 45trTˆtrBˆ + 15trTˆtrLˆ + 4 2 28α2 13α α 25648α2 4α α 500α2 + n 1 + 1 2 + 2 1 3 + 52α α + 3 G − 15 5 27 − 15 2 3 3 44α2 1660α2 176α2 + n2 1 2 3 (23) G − 45 − 27 − 9 and α2 16n β = 3 44 + G 3 2 − 3 (4π) α2 22α 304α + 3 408α 8trTˆ 8trBˆ + n 1 +6α + 3 3 − 3 − − G 15 2 3 (4π) α2 101α trTˆ 93α trTˆ 89α trBˆ 93α trBˆ + 3 5714α2 1 2 160α trTˆ 1 2 4 − 3 − 10 − 2 − 3 − 10 − 2 (4π) 160α trBˆ + 18trBˆ2 + 42(trBˆ)2 + 14trBˆtrLˆ 12trTˆBˆ + 18trTˆ2 + 84trTˆtrBˆ − 3 − + 14trTˆtrLˆ + 42(trTˆ)2 13α2 α α 241α2 308α α 20132α2 + n 1 1 2 + 2 + 1 3 + 28α α + 3 G − 30 − 10 6 45 2 3 9 242α2 22α2 2600α2 + n2 1 2 3 . (24) G − 135 − 3 − 27

In the above formulas nG denotes the number of fermion generations. It is obtained by labeling the closed quark and lepton loops present in the diagrams. To obtain the results for the three-loop gauge beta functions, one also needs the one-loop beta functions of the Yukawa couplings, cf. Eq. (6). They can be found in the literature, of course. Nevertheless we decided to re-calculate them as an additional check of our setup. For completeness, we present the analytical two-loop expressions which read

αt αt 17 βα = ǫ + 6αb +6αt + 4trLˆ + 12trBˆ + 12trTˆ α1 9α2 32α3 t − π π 2 − − 5 − − (4 ) α 9α2 9α α 76α α 1616α2 116α2 + t 1 1 2 35α2 + 1 3 + 36α α 3 + n 1 3 50 − 5 − 2 15 2 3 − 3 G 45 (4π) 320α2 393α α 225α α +4α2 + 3 + 24λˆ2 + 1 t + 2 t + 144α α 48λαˆ 48α2 2 9 20 4 3 t − t − t 15

7α α 99α α 15α α 15α α + 1 b + 2 b + 16α α 11α α α2 + 1 τ + 2 τ 9α α 20 4 3 b − t b − b 2 2 − t τ +5α α 9α2 , (25) b τ − τ

αb αb βα = ǫ + 6αb 6αt + 4trLˆ + 12trBˆ + 12trTˆ α1 9α2 32α3 b − π (4π)2 − − − − n2 2 o αb 29α1 27α1α2 2 124α1α3 1616α3 + − 35α + + 36α2α3 π 3 50 − 5 − 2 15 − 3 (4 ) 4α2 320α2 91α α 99α α 237α α + n − 1 +4α2 + 3 + 24λˆ2 + 1 t + 2 t + 16α α α2 + 1 b G 45 2 9 20 4 3 t − t 20 225α α 15α α 15α α + 2 b + 144α α 48λαˆ 11α α 48α2 + 1 τ + 2 τ +5α α 4 3 b − b − t b − b 2 2 t τ 9α α 9α2 , (26) − b τ − τ and

ατ ατ βα = ǫ + 6ατ + 4trLˆ + 12trBˆ + 12trTˆ 9α1 9α2 τ − π (4π)2 − − n o α 51α2 27α α 44α2 17α α + τ 1 + 1 2 35α2 + n 1 +4α2 + 24λˆ2 + 1 t 3 50 5 − 2 G 5 2 2 (4π) 45α α 5α α 45α α + 2 t + 80α α 27α2 + 1 b + 2 b + 80α α +6α α 27α2 2 3 t − t 2 2 3 b t b − b 537α α 165α α + 1 τ + 2 τ 48λαˆ 27α α 27α α 12α2 . (27) 20 4 − τ − t τ − b τ − τ The one-loop results have been expressed in terms of Yukawa matrices since these expressions enter the three-loop beta functions. At two-loop order, however, we refrain from reconstructing the general expression which would require an extension of the rules given in Eq. (19). Our independent calculation of the two-loop Yukawa beta functions is also interesting as there is a discrepancy between [20] and [23] concerning the absence of terms proportional

to αbαtλˆ in Eqs. (25) and (26). We were able to conﬁrm the results in Ref. [23]. In Appendices A and B we provide the results for the renormalization constants which lead to the beta functions discussed in this Section.

IV. CHECKS

We successfully performed a number of consistency checks and compared our results with those already available in the literature. We describe these checks in detail in this Section. The consistency checks show that all computed renormalization constants are local, that the renormalization constants of the gauge couplings are gauge parameter independent and that the beta functions are finite. We also find that the beta functions calculated by con- sidering different vertices in Lorenz gauge agree among themselves and with the results of the computation in BFG. 16

In order to test that the program FeynArtsToQ2E correctly translates FeynArts model files into model files for QGRAF/q2e, we reproduced the beta function for the Higgs self- coupling to one-loop order and the beta functions for the top and bottom quark, and the tau lepton Yukawa couplings to two-loop order (cf. previous Section). We have even considered quantities within the Minimal Supersymmetric Standard Model, like the relation between the squark masses within one generation, which is quite involved in case electroweak interactions are kept non-zero. Furthermore, we find that the divergent loop corrections to the BBB vertex vanish in the Lorenz gauge, as expected since for this vertex no renormalization is required. We performed the latter check up to three-loop order. Another check consists in verifying that in the vertex diagrams no infra-red divergences are introduced although one external momentum is set to zero. We do not have to consider two-point functions since they are infra-red safe. One can avoid infra-red divergences by introducing a common mass for the internal particles. Afterwards, the resulting integrals are evaluated in the limit q2 m2 where q is the non-vanishing external momentum of the vertex diagrams. This is conveniently≫ done by applying the rules of asymptotic expansion [41] which are encoded in the program exp. The setup described in Section II C is particularly useful for this test since exp takes over the task of generating FORM code for all relevant sub-diagrams which can be up to 35 for some of the diagrams; this makes the calculation significantly more complex. In our case the asymptotic expansion either leads to massless two-point functions or massive vacuum integrals or a combination of both. The former are computed with the help of the package MINCER, for the latter the package MATAD is used. As a result one obtains a series in m2/q2 where the coefficients contain numbers and ln(m2/q2) terms. For our purpose it is sufficient to restrict ourselves to the term (m2/q2)0 and check that no logarithms appear in the final result. With this method we have explicitly checked that the W1W2W3 and three-gluon vertices are free from infra-red divergences. Since the results for the gauge coupling renormalization constants agree with the ones obtained from the other vertices also the latter are infra-red safe. Let us finally comment on the comparison of our findings with the literature. We have successfully compared our results for the two-loop gauge beta functions with [9] and the two-loop Yukawa beta functions with [23]. Even partial results for the three-loop gauge beta functions were available in the literature [16]. That paper comprises all hitherto known three-loop corrections in a general quantum field theory based on a single gauge group, however, the presentation of the results relies on a quite intricate notation. Its specification to the SM is straightforward, however, a bit tedious. For convenience, the translation rules needed to specify the notation of [16] to ours is given in Appendix D. After specifying the notation of Ref. [16] to ours we find complete agreement, taking into account the following modifications in Eq. (33) of [16]:4 The terms 7g2 tr Y aY¯ b tr Y¯ bY aC(R) − 12r 2 a b b c g tr Y Y¯ tr Y Y¯ C(S)ca + (28) 12r have to be “symmetrized”, so that they read 1 7g2 tr Y aY¯ b tr Y¯ bY aC(R) 7g2 tr Y aY¯ b tr Y bY¯ aC(R) 2 − 12r − 12r ! 4 We want to thank the authors of [16] for pointing out the issue of symmetrization and for assistance in deriving the translation rules. 17

2 a b b c 2 a b b c 1 g tr Y Y¯ tr Y Y¯ C(S)ca g tr Y Y¯ tr Y¯ Y C(S)ca + + . (29) 2 12r − 12r ! Furthermore, one has to correct the obvious misprint

5g4 tr C(R)2Y¯ aY b 5g4 tr C(R)2Y¯ aY a + + . (30) 12r → 12r

V. NUMERICAL ANALYSIS

In this Section we discuss the numerical eﬀect of the new contributions to the gauge beta functions. We solve the corresponding renormalization group equations of the gauge couplings numerically and take into account the contributions from the Yukawa couplings and the Higgs self-coupling to two-loops. As boundary conditions we choose

αMS (M )=0.0169225 0.0000039 , 1 Z ± αMS (M )=0.033735 0.000020 , 2 Z ± αMS (M )=0.1173 0.00069 , 3 Z ± MS αt (MZ )=0.07514 , MS αb (MZ )=0.00002064 , αMS (M )=8.077 10−6 , τ Z · 4πλˆ =0.13 , (31)

where the ﬁrst six entries correspond to experimentally determined values while the value for the Higgs coupling is determined assuming a Higgs boson with mass 125 GeV:

2 2 2 ˆ mH 125 GeV 4πλ = 2 0.13 . (32) 2v2 ≈ 2 √2 174 GeV2 ≈ · · Note that the values in Eq. (31) are given in the full SM, the top quark being not integrated out. For a description how these values are determined from the knowledge of their directly measured counterparts [40], we refer to [40, 42]. It is noteworthy that for all three gauge couplings the sum of all three-loop terms involving at least one of the couplings αb, ατ or λ leads to corrections which are less than 0.1% of the difference between the two- and three-loop prediction. In Fig. 3 the running of the couplings α1, α2 and α3, is shown from µ = MZ up to high energies. At this scale no difference between one, two and three loops is visible, all curves lie on top of each other. The differences between the loop orders can be seen in Fig. 4 which magnifies the in- tersection point between α1 and α2. There is a clear jump between the one- (dotted) and two-loop (dashed) prediction. The difference between two and three loops (solid curves) is significantly smaller which suggests that perturbation theory converges very well. The experimental uncertainties for α1(MZ ) and α2(MZ ) as given in Eq. (31) are reflected by the bands around the three-loop results. Defining the difference between the two- and 18

0.05 0.045 0.04

3 0.035 α , 2 0.03 α , 1

α 0.025 0.02 0.015 0.01 2 4 6 8 10 12 14 16 µ log10( /GeV)

FIG. 3. The running of the gauge couplings at three loops. The curve with the smallest initial value corresponds to α1, the middle curve to α2, and the curve with the highest initial value to α3.

three-loop result as theoretical uncertainty one observes that it is smaller than the experimental one, however, of the same order of magnitude. Without the new three-loop calculation performed in this paper the theory uncertainty is much larger than the experimental one. Also in the case of α3 perturbation theory seems to converge well. However, in contrast to α1 and α2 the experimental uncertainty turns out to be much larger than the theoretical uncertainty. This is not surprising as the relative experimental uncertainty of α3 at the electroweak scale is quite large compared to its electroweak counterparts. The relative experimental and theoretical uncertainty of α3 is plotted as a function of the renormalization scale in Fig. 5. Note that by construction we have that ∆α3/α3 theory approaches zero for µ M . | → Z Let us finally identify the numerically most important contributions. This is done by 16 running from µ = MZ to µ = 10 GeV and by comparing the contribution of an individual term to the total difference between the two- and three-loop prediction. Similar results are also obtained for lower scales. About 90% of the three-loop corrections to the running of the gauge couplings arises from only a few terms. In the case of α1 there is only one term which dominates, namely the one 2 2 2 2 of order α1α3. For α2 one has a contribution of +56% from the (α2α3) term, +13% from 3 4 O order α2α3 and +37% from (α2). All other terms contribute at most 5% and partly also O 4 cancel each other. Except for the term of (α2) all these terms are presented in this paper for the first time. O The beta function β3 is dominated by the strong corrections, however, large cancellations 4 3 2 2 2 2 between the α3 (+137%), α3α2 (+45%), α3αt (+28%) and α3α2 (+17%) terms on the one 19

-1 x 10 0.237 0.2368 0.2365

2 0.2363 α , 1 0.236 α 0.2358 0.2355 0.2353 0.235 12.98 13 13.02 13.04 13.06 13.08 µ log10( /GeV)

FIG. 4. The running of the electroweak gauge couplings in the SM. The lines with positive slope correspond to α1, the lines with negative slope to α2. The dotted, dashed and solid lines correspond to one-, two- and three-loop precision, respectively. The bands around the three-loop curves visualize the experimental uncertainty.

3 2 hand and the α3αt ( 112%) and α3α2αt ( 16%) on the hand are observed. It is worth − 5 − noting that the four-loop term of order αs amounts to 58% of the diﬀerence between the two- and three-loop predictions. This number has been− obtained by adding the four-loop QCD term [17, 18] to β3.

VI. CONCLUSIONS

In this paper we present three-loop results for renormalization constants that are used to compute the three SM gauge coupling beta functions, taking into account all contributions. We have checked that our expressions agree with all partial results present in the literature. Furthermore the two-loop corrections to the Yukawa couplings have been computed. We have performed the calculation using both Lorenz gauge within the unbroken phase of the SM and BFG in the broken phase. Our final result is valid for a generic flavour structure with an arbitrary CKM matrix. It is furthermore sufficiently general to consider a fourth generation of quarks and leptons. In order to perform the calculation in an automated way we have written an interface, FeynArtsToQ2E, between the package FeynArts and our chain of programs (QGRAF, q2e, exp, MATAD, MINCER) allowing to handle the (106) Feynman diagrams, which have to be considered in the course of the calculation, inO an effective way. Thus, we could perform several checks involving various different Green’s functions. FeynArtsToQ2E is not limited 20

-2 10

-3 10 3 α / 3

∆α -4 10

-5 10 2 4 6 8 10 12 14 16 µ log10( /GeV)

FIG. 5. Comparison of the relative experimental and theoretical uncertainty of α3. The theoretical uncertainty is given by the dashed line, the solid curve corresponds to the experimental one.

to the SM but can easily be used for extensions like supersymmetric models.

ACKNOWLEDGMENTS

We thank David Kunz for tests concerning FeynArtsToQ2E and Miko laj Misiak for sug- gesting to generalize the Yukawa structure of the ﬁnal result. This work is supported by DFG through SFB/TR 9 “Computational Particle Physics”.

Appendix A: Renormalization constants

In this Appendix we present analytical results for the renormalization constants Zα1 , Zα2 ,

Zα3 , ZB, ZW , ZG, ZH , ZcW , ZcG , ZcCW , ZcCG, ZWWW , ZGGG, ZHHW , where the definition of Zαi is given in Eq. (5) and the field renormalization is defined through

bare bare bare B = ZBB , W = ZW W, G = ZGG , bare bare bare cW = ZcW cW , cG = ZcG cG , H = ZH H . (A1)

B, W , G, cW and cG denote the gauge boson and ghost fields. The scalar field H is defined in Eq. (D5). The renormalization constants for the three-particle vertices are also defined in a multiplicative way. Some of the results listed below contain the gauge parameters ξB, ξW and ξG. They are 21 conveniently defined via the corresponding gauge boson propagator which is given by µν qµqν g + (1 ξX ) 2 Dµν (q)= i− − q , (A2) X q2

with X = B, W, G. Note that ξX = 1 corresponds to Feynman and ξX = 0 to Landau gauge. Our analytical results read α 1 1 4n α 1 α 4n α 16n2 α Z =1+ 1 + G + 1 1 + G 1 + G 1 α1 4π ǫ 10 3 2 ǫ2 100 15 9 (4π) 1 9α 9α 17trTˆ trBˆ 3trLˆ 19α 3α 22α + 1 + 2 + n 1 + 2 + 3 ǫ 100 20 − 20 − 4 − 4 G 30 10 15 α 1 α2 n α2 8n2 α2 64n3 α2 + 1 1 + G 1 + G 1 + G 1 3 ǫ3 1000 25 15 27 (4π) 1 21α2 9α α 43α2 17α trTˆ 51α trTˆ 34α trTˆ 7α trBˆ + 1 + 1 2 2 + 1 + 2 + 3 1 ǫ2 1000 100 − 40 240 80 15 − 240 3α trBˆ 2α trBˆ 33α trLˆ 9α trLˆ trBˆ2 (trBˆ)2 5trBˆtrLˆ + 2 + 3 + 1 + 2 16 3 80 16 − 8 − 4 − 6 3trLˆ2 (trLˆ)2 11trTˆBˆ 17trTˆ2 11trTˆtrBˆ 31trTˆtrLˆ 17(trTˆ)2 + − 8 − 4 20 − 40 − 10 − 30 − 20 77α2 63α α 31α2 22α α 242α2 34α trTˆ 2α trBˆ + n 1 + 1 2 2 + 1 3 3 1 1 2α trLˆ G 180 50 − 60 75 − 45 − 15 − 3 − 1 266α2 4α α 2α2 176α α 88α2 + n2 1 + 1 2 + 2 + 1 3 + 3 G 135 5 15 45 135 1 163α2 261α α 3401α2 9α λˆ 3α λˆ 3λˆ2 2827α trTˆ 157α trTˆ + 1 + 1 2 + 2 + 1 + 2 1 2 ǫ 8000 800 960 50 10 − 5 − 2400 − 32 29α3trTˆ 1267α1trBˆ 437α2trBˆ 17α3trBˆ 843α1trLˆ 543α2trLˆ − 15 − 2400 − 160 − 15 − 800 − 160 61trBˆ2 17(trBˆ)2 157trBˆtrLˆ 87trLˆ2 33(trLˆ)2 trTˆBˆ 113trTˆ2 + + + + + + + 80 40 60 80 40 8 80 59trTˆtrBˆ 199trTˆtrLˆ 101(trTˆ)2 + + + 20 60 40 58α2 7α α 83α2 137α α α α 275α2 + n 1 1 2 + 2 1 3 2 3 + 3 G − 225 − 300 90 − 675 − 15 27 209α2 11α2 484α2 + n2 1 2 3 , (A3) G − 405 − 45 − 405 α 1 43 4n α 1 1849α 172n α 16n2 α Z =1+ 2 + G + 2 2 G 2 + G 2 α2 4π ǫ − 6 3 2 ǫ2 36 − 9 9 (4π) 1 3α 259α 3trTˆ 3trBˆ trLˆ α 49α + 1 2 + n 1 + 2 +2α ǫ 20 − 12 − 4 − 4 − 4 G 10 6 3 α 1 79507α2 1849n α2 344n2 α2 64n3 α2 + 2 2 + G 2 G 2 + G 2 π 3 ǫ3 − 216 9 − 9 27 (4 ) 22

1 α2 43α α 77959α2 17α trTˆ 181α trTˆ α trBˆ + 1 1 2 + 2 + 1 + 2 +2α trTˆ + 1 ǫ2 200 − 20 216 80 16 3 16 181α trBˆ 3α trLˆ 181α trLˆ 3trBˆ2 3(trBˆ)2 trBˆtrLˆ + 2 +2α trBˆ + 1 + 2 16 3 16 48 − 8 − 4 − 2 (trLˆ)2 3trTˆBˆ 3trTˆ2 3trTˆtrBˆ trTˆtrLˆ 3(trTˆ)2 trLˆ2 + − 12 4 − 8 − 2 − 2 − 4 − 8 7α2 31α α 22001α2 86α α 22α2 2α trLˆ + n 1 1 2 2 2 3 3 2α trTˆ 2α trBˆ 2 G 100 − 30 − 108 − 3 − 3 − 2 − 2 − 3 2α2 4α α 686α2 16α α 8α2 + n2 1 + 1 2 + 2 + 2 3 + 3 G 45 15 27 3 9 1 163α2 187α α 667111α2 α λˆ α λˆ 593α trTˆ 243α trTˆ + 1 + 1 2 2 + 1 + 2 λˆ2 1 2 ǫ 4800 160 − 5184 10 2 − − 480 − 32 7α trTˆ 533α trBˆ 243α trBˆ 7α trBˆ 17α trLˆ 81α trLˆ 19trBˆ2 3 1 2 3 1 2 + − 3 − 480 − 32 − 3 − 32 − 32 16 15(trBˆ)2 5trBˆtrLˆ 19trLˆ2 5(trLˆ)2 9trTˆBˆ 19trTˆ2 15trTˆtrBˆ + + + + + + + 8 4 48 24 8 16 4 5trTˆtrLˆ 15(trTˆ)2 7α2 13α α 6412α2 α α 13α α 125α2 + + + n 1 + 1 2 + 2 1 3 + 2 3 + 3 4 8 G − 45 60 81 − 45 3 9 11α2 415α2 44α2 + n2 1 2 3 , (A4) G − 135 − 81 − 27

α 1 4n α 1 88n α 16n2 α Z =1+ 3 11 + G + 3 121α G 3 + G 3 α3 4π ǫ − 3 2 ǫ2 3 − 3 9 (4π) 1 11α 3α 38α + 51α trTˆ trBˆ + n 1 + 2 + 3 ǫ − 3 − − G 60 4 3 α 1 176n2 α2 64n3 α2 + 3 1331α2 + 484n α2 G 3 + G 3 3 ǫ3 − 3 G 3 − 3 27 (4π) 1 17α trTˆ 3α trTˆ 74α trTˆ α trBˆ 3α trBˆ 74α trBˆ + 1309α2 + 1 + 2 + 3 + 1 + 2 + 3 ǫ2 3 60 4 3 12 4 3 trBˆ2 trBˆtrLˆ trTˆ2 trTˆtrLˆ (trBˆ)2 + trTˆBˆ 2trTˆtrBˆ (trTˆ)2 − 2 − − 3 − 2 − − 3 − 11α2 43α2 121α α 33α α 4354α2 8α trTˆ 8α trBˆ + n 1 2 1 3 2 3 3 3 3 G 1800 − 24 − 30 − 2 − 9 − 3 − 3 11α2 α2 22α α 1064α2 + n2 1 + 2 + 1 3 +2α α + 3 G 135 3 45 2 3 27 1 2857α2 101α trTˆ 31α trTˆ 40α trTˆ 89α trBˆ 31α trBˆ 40α trBˆ + 3 1 2 3 1 2 3 ǫ − 6 − 120 − 8 − 3 − 120 − 8 − 3 3trBˆ2 7(trBˆ)2 7trBˆtrLˆ 3trTˆ2 7trTˆtrLˆ 7(trTˆ)2 + + + trTˆBˆ + + 7trTˆtrBˆ + + 2 2 6 − 2 6 2 23

13α2 α α 241α2 77α α 7α α 5033α2 + n 1 1 2 + 2 + 1 3 + 2 3 + 3 G − 360 − 120 72 135 3 27 121α2 11α2 650α2 + n2 1 2 3 , (A5) G − 810 − 18 − 81

α 1 1 4n α 1 9α 9α 17trTˆ trBˆ 3trLˆ Z =1+ 1 G + 1 1 2 + + + B 4π ǫ − 10 − 3 2 ǫ − 100 − 20 20 4 4 (4π) 19α 3α 22α + n 1 2 3 G − 30 − 10 − 15 α 1 3α2 43α2 289α trTˆ 51α trTˆ 34α trTˆ α trBˆ + 1 1 + 2 1 2 3 1 π 3 ǫ2 − 1000 40 − 1200 − 80 − 15 − 48 (4 ) 3α trBˆ 2α trBˆ 9α trLˆ 9α trLˆ trBˆ2 (trBˆ)2 5trBˆtrLˆ 3trLˆ2 2 3 1 2 + + + + − 16 − 3 − 16 − 16 8 4 6 8 (trLˆ)2 11trTˆBˆ 17trTˆ2 11trTˆtrBˆ 31trTˆtrLˆ 17(trTˆ)2 + + + + + 4 − 20 40 10 30 20 11α2 31α2 242α2 38α2 2α2 88α2 + n 1 + 2 + 3 + n2 1 2 3 G − 180 60 45 G − 135 − 15 − 135 1 163α2 261α α 3401α2 9α λˆ 3α λˆ 3λˆ2 + 1 1 2 2 1 2 + ǫ − 8000 − 800 − 960 − 50 − 10 5 2827α trTˆ 157α trTˆ 29α trTˆ 1267α trBˆ 437α trBˆ 17α trBˆ + 1 + 2 + 3 + 1 + 2 + 3 2400 32 15 2400 160 15 843α trLˆ 543α trLˆ 61trBˆ2 17(trBˆ)2 157trBˆtrLˆ 87trLˆ2 + 1 + 2 800 160 − 80 − 40 − 60 − 80 33(trLˆ)2 trTˆBˆ 113trTˆ2 59trTˆtrBˆ 199trTˆtrLˆ 101(trTˆ)2 − 40 − 8 − 80 − 20 − 60 − 40 58α2 7α α 83α2 137α α α α 275α2 + n 1 + 1 2 2 + 1 3 + 2 3 3 G 225 300 − 90 675 15 − 27 209α2 11α2 484α2 + n2 1 + 2 + 3 , (A6) G 405 45 405

α 1 25 4n Z =1+ 2 G ξ W 4π ǫ 6 − 3 − W α 1 25α 8ξ α 4ξ α + 2 2 W 2 + ξ2 α + n 2α + W 2 2 ǫ2 − 4 − 3 W 2 G 2 3 (4π) 1 3α 113α 3trTˆ 3trBˆ trLˆ 11ξ α ξ2 α + 1 + 2 + + + W 2 W 2 ǫ − 20 8 4 4 4 − 4 − 2 α 13α + n 1 2 2α G − 10 − 2 − 3 α 1 1525α2 8n2 α2 89ξ α2 7ξ2 α2 + 2 2 + G 2 + W 2 + W 2 ξ3 α2 π 3 ǫ3 72 9 12 6 − W 2 (4 ) 24

86α2 10ξ α2 4ξ2 α2 + n 2 W 2 W 2 G − 9 − 3 − 3 1 α2 3α α 29629α2 17α trTˆ 21α trTˆ + 1 + 1 2 2 1 2 2α trTˆ ǫ2 − 200 20 − 432 − 80 − 16 − 3 α trBˆ 21α trBˆ 3α trLˆ 7α trLˆ 3trBˆ2 3(trBˆ)2 1 2 2α trBˆ 1 2 + + − 16 − 16 − 3 − 16 − 16 8 4 trBˆtrLˆ trLˆ2 (trLˆ)2 3trTˆBˆ 3trTˆ2 3trTˆtrBˆ trTˆtrLˆ 3(trTˆ)2 + + + + + + + 2 8 12 − 4 8 2 2 4 3α α 271α2 3α trTˆ 3α trBˆ α trLˆ 53ξ2 α2 7ξ3 α2 + ξ 1 2 2 2 2 2 + W 2 + W 2 W 20 − 24 − 4 − 4 − 4 12 6 7α2 α α 4273α2 22α2 + n 1 + 1 2 + 2 +2α α + 3 G − 100 10 108 2 3 3 α α 47α2 2ξ2 α2 2α2 118α2 8α2 + ξ 1 2 + 2 +2α α + W 2 + n2 1 2 3 W 10 6 2 3 3 G − 45 − 27 − 9 1 163α2 11α α 3ζ α α 143537α2 ζ α2 α λˆ α λˆ + 1 1 2 3 1 2 + 2 + 3 2 1 2 + λˆ2 ǫ − 4800 − 32 − 10 1728 2 − 10 − 2 593α trTˆ 79α trTˆ 7α trTˆ 533α trBˆ 79α trBˆ 7α trBˆ 17α trLˆ + 1 + 2 + 3 + 1 + 2 + 3 + 1 480 32 3 480 32 3 32 79α trLˆ 19trBˆ2 15(trBˆ)2 5trBˆtrLˆ 19trLˆ2 5(trLˆ)2 9trTˆBˆ + 2 96 − 16 − 8 − 4 − 48 − 24 − 8 19trTˆ2 15trTˆtrBˆ 5trTˆtrLˆ 15(trTˆ)2 − 16 − 4 − 4 − 8 105α2 11α2 ζ α2 7ξ3 α2 + ξ 2 2ζ α2 + ξ2 2 3 2 W 2 W − 8 − 3 2 W − 4 − 2 − 12 7α2 8α α 4ζ α α 7025α2 α α 32α α + n 1 + 1 2 3 1 2 2 + 12ζ α2 + 1 3 + 2 3 G 45 15 − 5 − 108 3 2 45 3 125α2 8ξ α2 11α2 185α2 44α2 16ζ α α 3 + W 2 + n2 1 + 2 + 3 , (A7) − 3 2 3 − 9 3 G 135 27 27 α 1 13 4n 3ξ Z =1+ 3 G G G 4π ǫ 2 − 3 − 2 α 1 117α 51ξ α 9ξ2 α + 3 3 G 3 + G 3 + n 3α +2ξ α 2 ǫ2 − 8 − 8 4 G 3 G 3 (4π) 1 531α 99ξ α 9ξ2 α 11α 3α 61α + + 3 + trTˆ + trBˆ G 3 G 3 + n 1 2 3 ǫ 16 − 16 − 8 G − 60 − 4 − 6 α 1 1209α2 4n2 α2 423ξ α2 9ξ2 α2 27ξ3 α2 + 3 3 + G 3 + G 3 + G 3 G 3 3 ǫ3 16 3 16 2 − 8 (4π) 15ξ α2 + n 22α2 G 3 3ξ2 α2 G − 3 − 2 − G 3 1 7957α2 17α trTˆ 3α trTˆ 25α trTˆ α trBˆ 3α trBˆ 25α trBˆ + 3 1 2 3 1 2 3 ǫ2 − 32 − 60 − 4 − 6 − 12 − 4 − 6 25

trBˆ2 trBˆtrLˆ trTˆ2 trTˆtrLˆ + + (trBˆ)2 + trTˆBˆ + + 2trTˆtrBˆ + + (trTˆ)2 2 3 − 2 3 1287α2 3α trTˆ 3α trBˆ 117ξ2 α2 63ξ3 α2 + ξ 3 3 3 + G 3 + G 3 G − 32 − 2 − 2 8 16 11α2 43α2 11α α 9α α 1691α2 + n 1 + 2 + 1 3 + 2 3 + 3 G − 1800 24 40 8 18 11α α 9α α 73α2 3ξ2 α2 11α2 α2 182α2 + ξ 1 3 + 2 3 + 3 + G 3 + n2 1 2 3 G 40 8 4 2 G − 135 − 3 − 27 1 9965α2 81ζ α2 101α trTˆ 31α trTˆ 91α trTˆ 89α trBˆ 31α trBˆ + 3 3 3 + 1 + 2 + 3 + 1 + 2 ǫ 32 − 16 120 8 12 120 8 91α trBˆ 3trBˆ2 7(trBˆ)2 7trBˆtrLˆ 3trTˆ2 7trTˆtrLˆ + 3 + trTˆBˆ 7trTˆtrBˆ 12 − 2 − 2 − 6 − 2 − − 6 7(trTˆ)2 1503α2 27ζ α2 297α2 27ζ α2 63ξ3 α2 + ξ 3 3 3 + ξ2 3 3 3 G 3 − 2 G − 32 − 4 G − 32 − 16 − 32 13α2 α α 241α2 3223α α 11ζ α α 293α α + n 1 + 1 2 2 + 1 3 3 1 3 + 2 3 9ζ α α G 360 120 − 72 2160 − 5 48 − 3 2 3 8155α2 121α2 11α2 860α2 3 + 22ζ α2 +6ξ α2 + n2 1 + 2 + 3 , (A8) − 54 3 3 G 3 G 810 18 81

α 1 3 ξ Z =1+ 2 W cW 4π ǫ 2 − 2 α 1 17α 3ξ2 α 1 179α 5n α ξ α + 2 2 + n α + W 2 + 2 G 2 + W 2 2 ǫ2 − 4 G 2 8 ǫ 48 − 6 8 (4π) α 1 1309α2 8n2 α2 17ξ α2 9ξ2 α2 5ξ3 α2 + 2 2 + G 2 + W 2 W 2 W 2 3 ǫ3 72 9 24 − 16 − 16 (4π) 145α2 ξ α2 1 3α α 29209α2 20n2 α2 3α trTˆ 3α trBˆ + n 2 W 2 + 1 2 2 G 2 2 2 G − 18 − 6 ǫ2 20 − 864 − 27 − 4 − 4 α trLˆ 37ξ α2 13ξ2 α2 ξ3 α2 α α 1535α2 ξ α2 2 + W 2 + W 2 + W 2 + n 1 2 + 2 +2α α W 2 − 4 96 16 6 G 10 108 2 3 − 12 1 33α α 3ζ α α 59125α2 70n2 α2 ζ α2 41α trTˆ 41α trBˆ + 1 2 + 3 1 2 + 2 G 2 3 2 + 2 + 2 ǫ − 80 20 2592 − 81 − 4 16 16 41α trLˆ 29α2 α2 ζ α2 ξ3 α2 + 2 + ξ 2 + ζ α2 + ξ2 2 + 3 2 W 2 48 W − 24 3 2 W − 4 4 − 8 3α α 2ζ α α 4573α2 15α α 7ξ α2 + n 1 2 + 3 1 2 2 6ζ α2 2 3 +8ζ α α + W 2 , G − 8 5 − 648 − 3 2 − 2 3 2 3 6 (A9)

α 1 9 3ξ Z =1+ 3 G cG 4π ǫ 4 − 4 26

α 1 315α 3n α 27ξ2 α 1 285α 5n α 9ξ α + 3 3 + G 3 + G 3 + 3 G 3 + G 3 2 ǫ2 − 32 2 32 ǫ 32 − 4 32 (4π) α 1 8295α2 4n2 α2 315ξ α2 243ξ2 α2 135ξ3 α2 + 3 3 + G 3 + G 3 G 3 G 3 π 3 ǫ3 128 3 128 − 128 − 128 (4 ) 149α2 3ξ α2 1 15587α2 10n2 α2 3α trTˆ 3α trBˆ + n 3 G 3 + 3 G 3 3 3 G − 8 − 8 ǫ2 − 128 − 9 − 2 − 2 45ξ α2 351ξ2 α2 9ξ3 α2 11α α 9α α 1597α2 3ξ α2 + G 3 + G 3 + G 3 + n 1 3 + 2 3 + 3 G 3 32 128 16 G 40 8 48 − 16 1 15817α2 35n2 α2 81ζ α2 23α trTˆ 23α trBˆ + 3 G 3 + 3 3 + 3 + 3 ǫ 192 − 27 32 8 8 153α2 27ζ α2 27α2 27ζ α2 27ξ3 α2 + ξ 3 + 3 3 + ξ2 3 + 3 3 G 3 G − 32 8 G − 32 32 − 64 33α α 11ζ α α 135α α 9ζ α α 637α2 21ξ α2 + n 1 3 + 3 1 3 2 3 + 3 2 3 3 + G 3 11ζ α2 , G − 32 10 − 32 2 − 36 8 − 3 3 (A10)

1 1 9α 9α 3ξ α 3ξ α Z =1+ 1 + 2 3trTˆ 3trBˆ trLˆ B 1 W 2 H 4π ǫ 20 4 − − − − 20 − 4 1 1 99α2 81α α 177α2 3α trTˆ 27α trTˆ 39α trBˆ + 1 + 1 2 2 1 2 + 12α trTˆ 1 2 ǫ2 800 80 − 32 − 40 − 8 3 − 40 (4π) 27α trBˆ 27α trLˆ 9α trLˆ 9trBˆ2 3trLˆ2 9trTˆBˆ 9trTˆ2 2 + 12α trBˆ + 1 2 + − 8 3 40 − 8 − 4 − 4 2 − 4 27α2 27α α 9ξ α α 9α trTˆ 9α trBˆ 3α trLˆ 9ξ2 α2 + ξ 1 1 2 + W 1 2 + 1 + 1 + 1 + B 1 B − 400 − 80 80 20 20 20 800 27α α 9α2 9α trTˆ 9α trBˆ 3α trLˆ 21ξ2 α2 + ξ 1 2 2 + 2 + 2 + 2 + W 2 W − 80 − 16 4 4 4 32 3α2 3α2 + n 1 + 2 G 10 2 1 93α2 27α α 511α2 17α trTˆ 45α trTˆ 5α trBˆ + 1 1 2 + 2 3λˆ2 1 2 10α trTˆ 1 ǫ − 1600 − 160 64 − − 16 − 16 − 3 − 16 45α trBˆ 15α trLˆ 15α trLˆ 27trBˆ2 9trLˆ2 3trTˆBˆ 27trTˆ2 2 10α trBˆ 1 2 + + + − 16 − 3 − 16 − 16 8 8 − 4 8 3ξ α2 3ξ2 α2 α2 5α2 W 2 W 2 + n 1 2 − 2 − 16 G − 4 − 4 1 1 429α3 891α2α 1593α α2 8555α3 93α2trTˆ 9α α trTˆ + 1 + 1 2 1 2 + 2 1 1 2 3 ǫ3 16000 3200 − 640 384 − 800 − 16 (4π) 435α2trTˆ 7α α trTˆ 177α2trBˆ 99α α trBˆ + 2 1 3 +9α α trTˆ 76α2trTˆ 1 1 2 32 − 5 2 3 − 3 − 800 − 80 435α2trBˆ 17α α trBˆ 339α2trLˆ 27α α trLˆ + 2 + 1 3 +9α α trBˆ 76α2trBˆ 1 + 1 2 32 5 2 3 − 3 − 800 80 27

145α2trLˆ 9trBˆ3 9α trBˆ2 9trTˆtrBˆ2 9trBˆtrBˆ2 3trLˆtrBˆ2 + 2 1 + 18α trBˆ2 32 − 4 − 20 3 − 4 − 4 − 4 3trLˆ3 27α trLˆ2 3trTˆtrLˆ2 3trBˆtrLˆ2 trLˆtrLˆ2 9trTˆ2Bˆ 9trTˆ3 + 1 + − 4 20 − 4 − 4 − 4 4 − 4 9α trTˆBˆ 9trTˆtrTˆBˆ 9trBˆtrTˆBˆ 3trLˆtrTˆBˆ 9trTˆBˆ2 1 36α trTˆBˆ + + + + − 20 − 3 2 2 2 4 9α trTˆ2 9trTˆtrTˆ2 9trBˆtrTˆ2 3trLˆtrTˆ2 + 1 + 18α trTˆ2 10 3 − 4 − 4 − 4 297α3 243α2α 531α α2 63ξ2 α α2 9α2trTˆ 81α α trTˆ + ξ 1 1 2 + 1 2 W 1 2 + 1 + 1 2 B − 16000 − 1600 640 − 640 800 160 9α α trTˆ 117α2trBˆ 81α α trBˆ 9α α trBˆ 81α2trLˆ 27α α trLˆ 1 3 + 1 + 1 2 1 3 1 + 1 2 − 5 800 160 − 5 − 800 160 27α trBˆ2 9α trLˆ2 27α trTˆBˆ 27α trTˆ2 + 1 + 1 1 + 1 80 80 − 40 80 81α2α 27α α2 27α α trTˆ 27α α trBˆ 9α α trLˆ + ξ 1 2 + 1 2 1 2 1 2 1 2 W 1600 320 − 80 − 80 − 80 297α2α 81α α2 367α3 9α α trTˆ 27α2trTˆ + ξ 1 2 1 2 + 2 + 1 2 2 9α α trTˆ W − 3200 − 320 128 160 − 32 − 2 3 117α α trBˆ 27α2trBˆ 81α α trLˆ 9α2trLˆ 27α trBˆ2 + 1 2 2 9α α trBˆ 1 2 2 + 2 160 − 32 − 2 3 − 160 − 32 16 9α trLˆ2 27α trTˆBˆ 27α trTˆ2 + 2 2 + 2 16 − 8 16 81α3 81α2α 27ξ α2α 27α2trTˆ 27α2trBˆ 9α2trLˆ + ξ2 1 + 1 2 W 1 2 1 1 1 B 16000 3200 − 3200 − 800 − 800 − 800 189α α2 63α3 63α2trTˆ 63α2trBˆ 21α2trLˆ 9ξ3 α3 77ξ3 α3 + ξ2 1 2 2 2 2 2 B 1 W 2 W 640 − 128 − 32 − 32 − 32 − 16000 − 128 7α3 27α2α 27α α2 263α3 α2trTˆ 16α2trTˆ 11α2trBˆ + n 1 + 1 2 + 1 2 2 1 3α2trTˆ + 3 1 G 40 40 40 − 24 − 3 − 2 3 − 15 16α2trBˆ α2trLˆ 9α3 9α α2 3α2trBˆ + 3 + 1 α2trLˆ + ξ 1 1 2 − 2 3 5 − 2 B − 200 − 40 9α2α 5α3 4α3 4α3 + ξ 1 2 2 + n2 1 + 2 W − 40 − 8 G 15 3 1 97α3 99α2α 4917α α2 121093α3 27α2λˆ 9α α λˆ 9α2λˆ + 1 1 2 + 1 2 2 1 1 2 2 ǫ2 − 32000 − 1280 1280 − 2304 − 200 − 20 − 8 9α λˆ2 9α λˆ2 541α2trTˆ 177α α trTˆ 885α2trTˆ α α trTˆ + 1 + 2 24λˆ3 1 1 2 2 1 3 20 4 − − 1600 − 160 − 64 − 10 33α α trTˆ 191α2trBˆ 57α α trBˆ 885α2trBˆ 2 3 + 198α2trTˆ 9λˆ2trTˆ + 1 + 1 2 2 − 2 3 − 1600 160 − 64 49α α trBˆ 33α α trBˆ 3α2trLˆ 45α α trLˆ 1 3 2 3 + 198α2trBˆ 9λˆ2trBˆ 1 1 2 − 10 − 2 3 − − 1600 − 32 28

295α2trLˆ 15trBˆ3 123α trBˆ2 117α trBˆ2 2 3λˆ2trLˆ + 1 2 39α trBˆ2 + 18λˆtrBˆ2 − 64 − 8 − 80 − 16 − 3 81trBˆtrBˆ2 27trLˆtrBˆ2 5trLˆ3 261α trLˆ2 39α trLˆ2 + + + 1 2 +6λˆtrLˆ2 8 8 8 − 80 − 16 27trTˆtrLˆ2 27trBˆtrLˆ2 9trLˆtrLˆ2 15trTˆ3 43α trTˆBˆ + + + + + 1 8 8 8 8 20 9α trTˆBˆ 11trLˆtrTˆBˆ 297α trTˆ2 + 2 + 46α trTˆBˆ 1 8 3 − 4 − 80 117α trTˆ2 81trTˆtrTˆ2 27trLˆtrTˆ2 2 39α trTˆ2 + 18λˆtrTˆ2 + + − 16 − 3 8 8 279α3 81α2α 1533α α2 9ξ α α2 9ξ2 α α2 9α λˆ2 51α2trTˆ + ξ 1 + 1 2 1 2 + W 1 2 + W 1 2 + 1 + 1 B 32000 3200 − 1280 40 320 20 320 27α α trTˆ 3α α trTˆ 3α2trBˆ 27α α trBˆ 3α α trBˆ 9α2trLˆ + 1 2 + 1 3 + 1 + 1 2 + 1 3 + 1 64 2 64 64 2 64 9α α trLˆ 81α trBˆ2 27α trLˆ2 9α trTˆBˆ 81α trTˆ2 + 1 2 1 1 + 1 1 64 − 160 − 160 80 − 160 279α2α 351α α2 141α3 9α λˆ2 51α α trTˆ 423α2trTˆ + ξ 1 2 1 2 2 + 2 + 1 2 + 2 W 6400 − 640 − 256 4 64 64 15α α trTˆ 15α α trBˆ 423α2trBˆ 15α α trBˆ 45α α trLˆ 141α2trLˆ + 2 3 + 1 2 + 2 + 2 3 + 1 2 + 2 2 64 64 2 64 64 81α trBˆ2 27α trLˆ2 9α trTˆBˆ 81α trTˆ2 2 2 + 2 2 − 32 − 32 16 − 32 27α α2 189α3 9α2trTˆ 9α2trBˆ 3α2trLˆ 33ξ3 α3 + ξ2 1 2 + 2 + 2 + 2 + 2 + W 2 W − 320 64 16 16 16 64 11α3 219α2α 39α α2 3241α3 11α2α 11α2trTˆ 3α2trTˆ + n 1 1 2 1 2 + 2 + 1 3 +3α2α 1 + 2 G 1200 − 400 − 80 144 25 2 3 − 30 2 40α2trTˆ 19α2trBˆ 3α2trBˆ 40α2trBˆ 9α2trLˆ α2trLˆ 3α3 3α α2 3 + 1 + 2 3 1 + 2 + ξ 1 + 1 2 − 3 30 2 − 3 − 10 2 B 80 16 3α2α 9α3 2α3 10α3 + ξ 1 2 2 + n2 1 2 3α α2 3α2α W 16 − 16 G − 9 − 9 − b t − b t 1 413α3 27ζ α3 279α2α 27ζ α2α 51α α2 9ζ α α2 70519α3 + 1 + 3 1 1 2 3 1 2 1 2 + 3 1 2 + 2 ǫ − 6000 2000 − 800 − 400 − 64 80 1728 69ζ α3 117α2λˆ 27ζ α2λˆ 39α α λˆ 9ζ α α λˆ 39α2λˆ 9ζ α2λˆ + 3 2 1 + 3 1 1 2 + 3 1 2 2 + 3 2 16 − 400 50 − 40 5 − 16 2 52831α2trTˆ ζ α2trTˆ 371α α trTˆ 3α λˆ2 15α λˆ2 + 12λˆ3 + 1 3 1 1 2 − 1 − 2 28800 − 100 − 320 27ζ α α trTˆ 2433α2trTˆ 63ζ α2trTˆ 2419α α trTˆ 68ζ α α trTˆ 3 1 2 2 + 3 2 + 1 3 3 1 3 − 10 − 128 4 180 − 5 163α α trTˆ 910α2trTˆ 45λˆ2trTˆ 27α (trTˆ)2 + 2 3 36ζ α α trTˆ 3 +8ζ α2trTˆ + + 1 4 − 3 2 3 − 9 3 3 2 20 29

27α (trTˆ)2 5479α2trBˆ 29ζ α2trBˆ 671α α trBˆ 9ζ α α trBˆ + 2 + 1 + 3 1 1 2 + 3 1 2 4 28800 100 − 320 5 2433α2trBˆ 63ζ α2trBˆ 991α α trBˆ 163α α trBˆ 2 + 3 2 + 1 3 4ζ α α trBˆ + 2 3 − 128 4 180 − 3 1 3 4 910α2trBˆ 45λˆ2trBˆ 27α trTˆtrBˆ 27α trTˆtrBˆ 36ζ α α trBˆ 3 +8ζ α2trBˆ + + 1 + 2 − 3 2 3 − 9 3 3 2 10 2 27α (trBˆ)2 27α (trBˆ)2 8517α2trLˆ 117ζ α2trLˆ 411α α trLˆ 18ζ α α trLˆ + 1 + 2 + 1 3 1 + 1 2 3 1 2 20 4 3200 − 100 320 − 5 811α2trLˆ 21ζ α2trLˆ 15λˆ2trLˆ 9α trTˆtrLˆ 9α trTˆtrLˆ 9α trBˆtrLˆ 2 + 3 2 + + 1 + 2 + 1 − 128 4 2 10 2 10 9α trBˆtrLˆ 3α (trLˆ)2 3α (trLˆ)2 25trBˆ3 303α trBˆ2 9ζ α trBˆ2 + 2 + 1 + 2 + 3ζ trBˆ3 + 1 3 1 2 20 4 16 − 3 80 − 5 279α trBˆ2 5α trBˆ2 + 2 9ζ α trBˆ2 3 + 24ζ α trBˆ2 15λˆtrBˆ2 16 − 3 2 − 2 3 3 − 25trLˆ3 33α trLˆ2 9ζ α trLˆ2 93α trLˆ2 18trBˆtrBˆ2 6trLˆtrBˆ2 + ζ trLˆ3 + 1 + 3 1 + 2 − − 48 − 3 80 5 16 25trTˆ3 3ζ α trLˆ2 5λˆtrLˆ2 6trTˆtrLˆ2 6trBˆtrLˆ2 2trLˆtrLˆ2 + − 3 2 − − − − 16 31α trTˆBˆ 8ζ α trTˆBˆ 21α trTˆBˆ 3ζ trTˆ3 + 1 3 1 + 2 19α trTˆBˆ + 16ζ α trTˆBˆ − 3 40 − 5 8 − 3 3 3 trLˆtrTˆBˆ 211α trTˆ2 3ζ α trTˆ2 + + 1 + 3 1 2 80 5 279α trTˆ2 5α trTˆ2 + 2 9ζ α trTˆ2 3 + 24ζ α trTˆ2 15λˆtrTˆ2 18trTˆtrTˆ2 16 − 3 2 − 2 3 3 − − 507ξ α3 39α3 3ζ α3 5ξ3 α3 6trLˆtrTˆ2 W 2 + ξ2 2 3 2 W 2 − − 64 W − 32 − 8 − 16 158α3 19ζ α3 3α2α 9ζ α2α 3α α2 3ζ α α2 2381α3 + n 1 + 3 1 1 2 + 3 1 2 1 2 + 3 1 2 2 G − 225 25 − 40 25 − 10 5 − 216 33α2α 44ζ α2α 45α2α 127α2trTˆ 21α2trTˆ 15ζ α3 1 3 + 3 1 3 2 3 + 12ζ α2α + 1 + 2 − 3 2 − 20 25 − 4 3 2 3 120 8 32α2trTˆ 31α2trBˆ 21α2trBˆ 32α2trBˆ 39α2trLˆ 7α2trLˆ 17ξ α3 + 3 + 1 + 2 + 3 + 1 + 2 + W 2 3 120 8 3 40 8 8 7α3 35α3 277α α2 277α α2 + n2 1 2 t b b t , (A11) G − 27 − 27 − 16 − 16

ξW α2 1 ξW α2 1 5α2 ξW α2 1 3α2 ZcCW =1 + + + ξW α2 − 4π ǫ π 2 ǫ − 4 − 4 ǫ2 2 (4 ) ξ α 1 61α2 2n α2 + W 2 2 + G 2 3ξ α2 ξ2 α2 3 ǫ3 − 12 3 − W 2 − W 2 (4π) 1 221α2 5n α2 3ξ2 α2 + 2 G 2 +4ξ α2 + W 2 ǫ2 24 − 3 W 2 4 30

1 373α2 5n α2 13ξ α2 5ξ2 α2 + 2 + G 2 W 2 W 2 , (A12) ǫ − 48 2 − 8 − 12

3 ξGα3 1 ξGα3 1 27α3 9ξGα3 1 45α3 9ξGα3 ZcCG =1 + + + − 2 4π ǫ π 2 ǫ2 8 4 ǫ − 16 − 16 (4 ) ξ α 1 279α2 3n α2 81ξ α2 27ξ2 α2 + G 3 3 + G 3 G 3 G 3 3 ǫ3 − 16 2 − 8 − 8 (4π) 1 1035α2 15n α2 27ξ α2 81ξ2 α2 + 3 G 3 + G 3 + G 3 ǫ2 32 − 4 2 32 1 1809α2 45n α2 351ξ α2 45ξ2 α2 + 3 + G 3 G 3 G 3 , (A13) ǫ − 64 8 − 64 − 32

1 α 8 4n 3ξ Z =1+ 2 G W WWW ǫ 4π 3 − 3 − 2 α 1 7ξ α 15ξ2 α + 2 6α W 2 + W 2 + n 3α +2ξ α 2 ǫ2 − 2 − 4 8 G 2 W 2 (4π) 1 3α 499α 3trTˆ 3trBˆ trLˆ 33ξ α 3ξ2 α + 1 + 2 + + + W 2 W 2 ǫ − 20 48 4 4 4 − 8 − 4 α 17α + n 1 2 2α G − 10 − 3 − 3 α 1 70α2 4n2 α2 59ξ α2 5ξ2 α2 35ξ3 α2 + 2 2 + G 2 + W 2 W 2 W 2 3 ǫ3 3 3 8 − 8 − 16 (4π) 43α2 13ξ α2 5ξ2 α2 + n 2 W 2 W 2 G − 3 − 2 − 2 1 α2 9α α 26057α2 17α trTˆ 27α trTˆ α trBˆ + 1 + 1 2 2 1 2 2α trTˆ 1 ǫ2 − 200 40 − 432 − 80 − 16 − 3 − 16 27α trBˆ 3α trLˆ 9α trLˆ 3trBˆ2 3(trBˆ)2 trBˆtrLˆ trLˆ2 2 2α trBˆ 1 2 + + + + − 16 − 3 − 16 − 16 8 4 2 8 (trLˆ)2 3trTˆBˆ 3trTˆ2 3trTˆtrBˆ trTˆtrLˆ 3(trTˆ)2 + + + + + 12 − 4 8 2 2 4 9α α 165α2 9α trTˆ 9α trBˆ 3α trLˆ 157ξ2 α2 17ξ3 α2 + ξ 1 2 2 2 2 2 + W 2 + W 2 W 40 − 32 − 8 − 8 − 8 16 8 7α2 3α α 4433α2 22α2 + n 1 + 1 2 + 2 +3α α + 3 G − 100 20 108 2 3 3 3α α 21α2 2α2 128α2 8α2 + ξ 1 2 + 2 +3α α + ξ2 α2 + n2 1 2 3 W 20 2 2 3 W 2 G − 45 − 27 − 9 1 163α2 11α α 9ζ α α 312361α2 3ζ α2 α λˆ α λˆ + 1 + 1 2 3 1 2 + 2 + 3 2 1 2 + λˆ2 ǫ − 4800 160 − 20 5184 4 − 10 − 2 593α trTˆ 3α trTˆ 7α trTˆ 533α trBˆ 3α trBˆ 7α trBˆ 17α trLˆ + 1 2 + 3 + 1 2 + 3 + 1 480 − 32 3 480 − 32 3 32 31

2 2 2 2 α2trLˆ 19trBˆ 15(trBˆ) 5trBˆtrLˆ 19trLˆ 5(trLˆ) 9trTˆBˆ − 32 − 16 − 8 − 4 − 48 − 24 − 8 19trTˆ2 15trTˆtrBˆ 5trTˆtrLˆ 15(trTˆ)2 315α2 + ξ 2 3ζ α2 − 16 − 4 − 4 − 8 W − 16 − 3 2 33α2 3ζ α2 7ξ3 α2 7α2 109α α 6ζ α α + ξ2 2 3 2 W 2 + n 1 + 1 2 3 1 2 W − 8 − 4 − 8 G 45 120 − 5 37577α2 α α 109α α 125α2 2 + 18ζ α2 + 1 3 + 2 3 24ζ α α 3 +4ξ α2 − 648 3 2 45 6 − 3 2 3 − 9 W 2 11α2 625α2 44α2 + n2 1 + 2 + 3 , (A14) G 135 81 27

1 α 17 4n 9ξ Z =1+ 3 G G GGG ǫ 4π 4 − 3 − 4 α 1 459α 9ξ α 135ξ2 α 9α + 3 3 G 3 + G 3 + n 3 +3ξ α 2 ǫ2 − 32 − 2 32 G 2 G 3 (4π) 1 777α 297ξ α 27ξ2 α 11α 3α 107α + 3 + trTˆ + trBˆ G 3 G 3 + n 1 2 3 ǫ 32 − 32 − 16 G − 60 − 4 − 12 α 1 10863α2 3537ξ α2 135ξ2 α2 945ξ3 α2 + 3 3 +2n2 α2 + G 3 G 3 G 3 π 3 ǫ3 128 G 3 128 − 128 − 128 (4 ) 117ξ α2 45ξ2 α2 + n 33α2 G 3 G 3 G − 3 − 8 − 8 1 28079α2 17α trTˆ 3α trTˆ 59α trTˆ α trBˆ 3α trBˆ 59α trBˆ + 3 1 2 3 1 2 3 ǫ2 − 128 − 60 − 4 − 12 − 12 − 4 − 12 trBˆ2 trBˆtrLˆ trTˆ2 trTˆtrLˆ + + (trBˆ)2 + trTˆBˆ + + 2trTˆtrBˆ + + (trTˆ)2 2 3 − 2 3 621α2 9α trTˆ 9α trBˆ 4185ξ2 α2 459ξ3 α2 + ξ 3 3 3 + G 3 + G 3 G − 32 − 4 − 4 128 64 11α2 43α2 33α α 27α α 14101α2 9ξ2 α2 + n 1 + 2 + 1 3 + 2 3 + 3 + G 3 G − 1800 24 80 16 144 4 33α α 27α α 393α2 11α2 α2 197α2 + ξ 1 3 + 2 3 + 3 + n2 1 2 3 G 80 16 16 G − 135 − 3 − 27 1 43973α2 243ζ α2 101α trTˆ 31α trTˆ 113α trTˆ 89α trBˆ 31α trBˆ + 3 3 3 + 1 + 2 + 3 + 1 + 2 ǫ 192 − 32 120 8 24 120 8 113α trBˆ 3trBˆ2 7(trBˆ)2 7trBˆtrLˆ 3trTˆ2 7trTˆtrLˆ + 3 + trTˆBˆ 7trTˆtrBˆ 24 − 2 − 2 − 6 − 2 − − 6 7(trTˆ)2 4509α2 81ζ α2 891α2 81ζ α2 189ξ3 α2 + ξ 3 3 3 + ξ2 3 3 3 G 3 − 2 G − 64 − 8 G − 64 − 32 − 64 13α2 α α 241α2 10901α α 33ζ α α 991α α 27ζ α α + n 1 + 1 2 2 + 1 3 3 1 3 + 2 3 3 2 3 G 360 120 − 72 4320 − 10 96 − 2 32

14399α2 121α2 11α2 965α2 3 +9ξ α2 + 33ζ α2 + n2 1 + 2 + 3 , (A15) − 108 G 3 3 3 G 810 18 81 1 1 9α 3α 3ξ α 5ξ α Z =1 + 1 + 2 3trTˆ 3trBˆ trLˆ B 1 W 2 HHW 4π ǫ 20 4 − − − − 20 − 4 1 1 99α2 27α α 77α2 3α trTˆ 9α trTˆ 39α trBˆ + 1 + 1 2 2 1 + 2 + 12α trTˆ 1 2 ǫ2 800 80 − 32 − 40 8 3 − 40 (4π) 9α trBˆ 27α trLˆ 3α trLˆ 9trBˆ2 3trLˆ2 9trTˆBˆ 9trTˆ2 + 2 + 12α trBˆ + 1 + 2 + 8 3 40 8 − 4 − 4 2 − 4 27α2 9α α 9α trTˆ 9α trBˆ 3α trLˆ 3ξ α α + ξ 1 1 2 + 1 + 1 + 1 + W 1 2 B − 400 − 80 20 20 20 16 9α α 15α2 15α trTˆ 15α trBˆ 5α trLˆ 9ξ2 α2 45ξ2 α2 + ξ 1 2 + 2 + 2 + 2 + 2 + B 1 + W 2 W − 16 16 4 4 4 800 32 3α2 α2 + n 1 + 2 G 10 2 1 93α2 27α α 817α2 17α trTˆ 45α trTˆ + 1 1 2 + 2 3λˆ2 1 2 10α trTˆ ǫ − 1600 − 160 192 − − 16 − 16 − 3 5α trBˆ 45α trBˆ 15α trLˆ 15α trLˆ 27trBˆ2 9trLˆ2 1 2 10α trBˆ 1 2 + + − 16 − 16 − 3 − 16 − 16 8 8 3trTˆBˆ 27trTˆ2 23ξ α2 7ξ2 α2 α2 5α2 + W 2 W 2 + n 1 2 − 4 8 − 8 − 16 G − 4 − 12 1 1 429α3 297α2α 693α α2 12551α3 93α2trTˆ 9α α trTˆ + 1 + 1 2 1 2 + 2 1 1 2 π 3 ǫ3 16000 3200 − 640 1152 − 800 − 20 (4 ) 27α2trTˆ 7α α trTˆ 177α2trBˆ 9α α trBˆ 2 1 3 9α α trTˆ 76α2trTˆ 1 + 1 2 − 32 − 5 − 2 3 − 3 − 800 40 27α2trBˆ 17α α trBˆ 339α2trLˆ 27α α trLˆ 2 + 1 3 9α α trBˆ 76α2trBˆ 1 1 2 − 32 5 − 2 3 − 3 − 800 − 40 9α2trLˆ 9trBˆ3 9α trBˆ2 27α trBˆ2 9trTˆtrBˆ2 2 1 + 2 + 18α trBˆ2 − 32 − 4 − 20 8 3 − 4 9trBˆtrBˆ2 3trLˆtrBˆ2 3trLˆ3 27α trLˆ2 9α trLˆ2 3trTˆtrLˆ2 + 1 + 2 − 4 − 4 − 4 20 8 − 4 3trBˆtrLˆ2 trLˆtrLˆ2 9trTˆ2Bˆ 9trTˆ3 9α trTˆBˆ 27α trTˆBˆ + 1 2 − 4 − 4 4 − 4 − 20 − 4 9trTˆtrTˆBˆ 9trBˆtrTˆBˆ 3trLˆtrTˆBˆ 9trTˆBˆ2 9α trTˆ2 36α trTˆBˆ + + + + + 1 − 3 2 2 2 4 10 27α trTˆ2 9trTˆtrTˆ2 9trBˆtrTˆ2 3trLˆtrTˆ2 + 2 + 18α trTˆ2 8 3 − 4 − 4 − 4 297α3 81α2α 231α α2 9α2trTˆ 27α α trTˆ + ξ 1 1 2 + 1 2 + 1 1 2 B − 16000 − 1600 640 800 − 160 9α α trTˆ 117α2trBˆ 27α α trBˆ 9α α trBˆ 81α2trLˆ 1 3 + 1 1 2 1 3 1 − 5 800 − 160 − 5 − 800 33

9α α trLˆ 27α trBˆ2 9α trLˆ2 27α trTˆBˆ 27α trTˆ2 1 2 + 1 + 1 1 + 1 − 160 80 80 − 40 80 27α2α 9α α2 9α α trTˆ 9α α trBˆ 3α α trLˆ 27ξ2 α α2 + ξ 1 2 1 2 1 2 1 2 1 2 W 1 2 W 320 − 64 − 16 − 16 − 16 − 128 99α2α 27α α2 745α3 3α α trTˆ 225α2trTˆ + ξ 1 2 + 1 2 2 + 1 2 2 15α α trTˆ W − 640 64 − 384 32 − 32 − 2 3 39α α trBˆ 225α2trBˆ 27α α trLˆ 75α2trLˆ 45α trBˆ2 + 1 2 2 15α α trBˆ 1 2 2 + 2 32 − 32 − 2 3 − 32 − 32 16 15α trLˆ2 45α trTˆBˆ 45α trTˆ2 81α3 27α2α + 2 2 + 2 + ξ2 1 + 1 2 16 − 8 16 B 16000 3200 2 ˆ 2 ˆ 2 ˆ 2 27α1trT 27α1trB 9α1trL 9ξW α1α2 − 800 − 800 − 800 − 640 81α α2 405α3 135α2trTˆ 135α2trBˆ 45α2trLˆ + ξ2 1 2 2 2 2 2 W 128 − 128 − 32 − 32 − 32 3 3 3 3 9ξBα1 195ξW α2 − 16000 − 128 7α3 9α2α 9α α2 317α3 α2trTˆ 16α2trTˆ 11α2trBˆ + n 1 + 1 2 + 1 2 2 1 + 3 1 G 40 40 40 − 72 − 3 3 − 15 16α2trBˆ α2trLˆ 9α3 3α α2 3α2α 5α3 + 3 + 1 + ξ 1 1 2 + ξ 1 2 + 2 3 5 B − 200 − 40 W − 8 24 4α3 4α3 + n2 1 + 2 G 15 9 1 97α3 63α2α 2901α α2 193057α3 27α2λˆ 9α α λˆ + 1 + 1 2 + 1 2 2 1 1 2 ǫ2 − 32000 6400 1280 − 6912 − 200 − 20 9α2λˆ 9α λˆ2 27α λˆ2 541α2trTˆ 39α α trTˆ 149α2trTˆ 2 + 1 + 2 24λˆ3 1 + 1 2 + 2 − 8 20 4 − − 1600 80 64 α α trTˆ 3α α trTˆ 191α2trBˆ 33α α trBˆ 1 3 2 3 + 198α2trTˆ 9λˆ2trTˆ + 1 + 1 2 − 10 − 2 3 − 1600 40 149α2trBˆ 49α α trBˆ 3α α trBˆ 3α2trLˆ + 2 1 3 2 3 + 198α2trBˆ 9λˆ2trBˆ 1 64 − 10 − 2 3 − − 1600 149α2trLˆ 15trBˆ3 123α trBˆ2 99α trBˆ2 + 2 3λˆ2trLˆ + 1 2 39α trBˆ2 192 − 8 − 80 − 8 − 3 81trBˆtrBˆ2 27trLˆtrBˆ2 5trLˆ3 261α trLˆ2 + 18λˆtrBˆ2 + + + 1 8 8 8 − 80 33α trLˆ2 27trTˆtrLˆ2 27trBˆtrLˆ2 9trLˆtrLˆ2 2 +6λˆtrLˆ2 + + + − 8 8 8 8 15trTˆ3 43α trTˆBˆ 9α trTˆBˆ 81trTˆtrTˆ2 27trLˆtrTˆ2 + + 1 + 2 + 46α trTˆBˆ + + 8 20 4 3 8 8 11trLˆtrTˆBˆ 297α trTˆ2 99α trTˆ2 1 2 39α trTˆ2 + 18λˆtrTˆ2 − 4 − 80 − 8 − 3 34

279α3 81α2α 817α α2 9α λˆ2 51α2trTˆ 27α α trTˆ + ξ 1 + 1 2 1 2 + 1 + 1 + 1 2 B 32000 3200 − 1280 20 320 64 3α α trTˆ 3α2trBˆ 27α α trBˆ 3α α trBˆ 9α2trLˆ 9α α trLˆ + 1 3 + 1 + 1 2 + 1 3 + 1 + 1 2 2 64 64 2 64 64 81α trBˆ2 27α trLˆ2 9α trTˆBˆ 81α trTˆ2 69ξ α α2 21ξ2 α α2 1 1 + 1 1 + W 1 2 + W 1 2 − 160 − 160 80 − 160 160 320 93α2α 693α α2 6515α3 15α λˆ2 85α α trTˆ 777α2trTˆ + ξ 1 2 1 2 + 2 + 2 + 1 2 + 2 W 1280 − 640 768 4 64 64 25α α trTˆ 25α α trBˆ 777α2trBˆ 25α α trBˆ 75α α trLˆ + 2 3 + 1 2 + 2 + 2 3 + 1 2 2 64 64 2 64 259α2trLˆ 135α trBˆ2 45α trLˆ2 15α trTˆBˆ 135α trTˆ2 + 2 2 2 + 2 2 64 − 32 − 32 16 − 32 63α α2 473α3 21α2trTˆ 21α2trBˆ 7α2trLˆ 241ξ3 α3 + ξ2 1 2 + 2 + 2 + 2 + 2 + W 2 W − 320 64 16 16 16 192 11α3 69α2α 17α α2 4123α3 11α2α 11α2trTˆ + n 1 1 2 1 2 + 2 + 1 3 + α2α 1 G 1200 − 400 − 80 432 25 2 3 − 30 40α2trTˆ 19α2trBˆ 40α2trBˆ 9α2trLˆ α2trLˆ α2trTˆ 3 + 1 α2trBˆ 3 1 2 − 2 − 3 30 − 2 − 3 − 10 − 3 3α3 α α2 5α2α 103α3 2α3 10α3 + ξ 1 + 1 2 + ξ 1 2 2 + n2 1 2 B 80 16 W 16 − 48 G − 9 − 27 3α α2 3α2α − t b − t b 1 413α3 27ζ α3 279α2α 27ζ α2α 123α α2 3ζ α α2 + 1 + 3 1 1 2 3 1 2 1 2 3 1 2 ǫ − 6000 2000 − 800 − 400 − 320 − 80 93307α3 73ζ α3 117α2λˆ 27ζ α2λˆ 39α α λˆ 9ζ α α λˆ 39α2λˆ + 2 + 3 2 1 + 3 1 1 2 + 3 1 2 2 5184 16 − 400 50 − 40 5 − 16 9ζ α2λˆ 52831α2trTˆ ζ α2trTˆ 371α α trTˆ + 3 2 3α λˆ2 15α λˆ2 + 12λˆ3 + 1 3 1 1 2 2 − 1 − 2 28800 − 100 − 320 27ζ α α trTˆ 2761α2trTˆ 63ζ α2trTˆ 2419α α trTˆ 68ζ α α trTˆ 3 1 2 2 + 3 2 + 1 3 3 1 3 − 10 − 128 4 180 − 5 163α α trTˆ 910α2trTˆ 45λˆ2trTˆ + 2 3 36ζ α α trTˆ 3 +8ζ α2trTˆ + 4 − 3 2 3 − 9 3 3 2 27α (trTˆ)2 27α (trTˆ)2 5479α2trBˆ 29ζ α2trBˆ 671α α trBˆ + 1 + 2 + 1 + 3 1 1 2 20 4 28800 100 − 320 9ζ α α trBˆ 2761α2trBˆ 63ζ α2trBˆ 991α α trBˆ + 3 1 2 2 + 3 2 + 1 3 4ζ α α trBˆ 5 − 128 4 180 − 3 1 3 163α α trBˆ 910α2trBˆ 45λˆ2trBˆ + 2 3 36ζ α α trBˆ 3 +8ζ α2trBˆ + 4 − 3 2 3 − 9 3 3 2 27α trTˆtrBˆ 27α trTˆtrBˆ 27α (trBˆ)2 27α (trBˆ)2 8517α2trLˆ + 1 + 2 + 1 + 2 + 1 10 2 20 4 3200 35

117ζ α2trLˆ 411α α trLˆ 18ζ α α trLˆ 2761α2trLˆ 21ζ α2trLˆ 3 1 + 1 2 3 1 2 2 + 3 2 − 100 320 − 5 − 384 4 15λˆ2trLˆ 9α trTˆtrLˆ 9α trTˆtrLˆ 9α trBˆtrLˆ 9α trBˆtrLˆ + + 1 + 2 + 1 + 2 2 10 2 10 2 3α (trLˆ)2 3α (trLˆ)2 25trBˆ3 303α trBˆ2 9ζ α trBˆ2 + 1 + 2 + 3ζ trBˆ3 + 1 3 1 20 4 16 − 3 80 − 5 279α trBˆ2 5α trBˆ2 + 2 9ζ α trBˆ2 3 + 24ζ α trBˆ2 15λˆtrBˆ2 16 − 3 2 − 2 3 3 − 25trLˆ3 33α trLˆ2 18trBˆtrBˆ2 6trLˆtrBˆ2 + ζ trLˆ3 + 1 − − 48 − 3 80 9ζ α trLˆ2 93α trLˆ2 + 3 1 + 2 3ζ α trLˆ2 5λˆtrLˆ2 6trTˆtrLˆ2 6trBˆtrLˆ2 5 16 − 3 2 − − − 25trTˆ3 31α trTˆBˆ 8ζ α trTˆBˆ 2trLˆtrLˆ2 + 3ζ trTˆ3 + 1 3 1 − 16 − 3 40 − 5 21α trTˆBˆ + 2 19α trTˆBˆ + 16ζ α trTˆBˆ 8 − 3 3 3 trLˆtrTˆBˆ 211α trTˆ2 3ζ α trTˆ2 279α trTˆ2 + + 1 + 3 1 + 2 9ζ α trTˆ2 2 80 5 16 − 3 2 5α trTˆ2 3 + 24ζ α trTˆ2 15λˆtrTˆ2 18trTˆtrTˆ2 6trLˆtrTˆ2 − 2 3 3 − − − 927α3 83α3 5ζ α3 29ξ3 α3 + ξ 2 ζ α3 + ξ2 2 3 2 W 2 W − 64 − 3 2 W − 32 − 8 − 48 158α3 19ζ α3 3α2α 9ζ α2α 3α α2 ζ α α2 1285α3 + n 1 + 3 1 1 2 + 3 1 2 + 1 2 + 3 1 2 2 G − 225 25 − 40 25 40 5 − 324 33α2α 44ζ α2α 15α2α 127α2trTˆ 21α2trTˆ 9ζ α3 1 3 + 3 1 3 2 3 +4ζ α2α + 1 + 2 − 3 2 − 20 25 − 4 3 2 3 120 8 32α2trTˆ 31α2trBˆ 21α2trBˆ 32α2trBˆ 39α2trLˆ 7α2trLˆ 83ξ α3 + 3 + 1 + 2 + 3 + 1 + 2 + W 2 3 120 8 3 40 8 24 7α3 35α3 277α α2 277α α2 + n2 1 2 t b b t . (A16) G − 27 − 81 − 16 − 16

Note that in the results for ZH and ZHHW there are terms which contain explicitly αb and αt since we have not been able to reconstruct the corresponding expressions in terms of ˆ ˆ B and T . They drop out in the ﬁnal result for Zα2 .

Appendix B: Two-loop Yukawa coupling renormalization constants

This Subsection contains the two-loop results for the Yukawa coupling renormalization constants deﬁned through

bare αi = Zαi αi , (B1) 36 with i = t, b, τ. They read 1 1 17α 9α 3α 3α Z =1+ 1 2 8α + t b + 3trTˆ + 3trBˆ + trLˆ αt 4π ǫ − 20 − 4 − 3 2 − 2 1 1 51α2 153α α 339α2 34α α 459α α + 1 + 1 2 + 2 + 1 3 + 18α α + 76α2 1 t 2 ǫ2 160 80 32 5 2 3 3 − 80 (4π) 243α α 81α2 117α α 81α α 45α α 9α2 2 t 54α α + t 1 b 2 b 18α α + t b + b − 16 − 3 t 4 − 80 − 16 − 3 b 4 2 79α α 27α α 33α α 15α α 7α2 1 τ 2 τ 8α α + t τ + b τ + τ − 40 − 8 − 3 τ 4 4 4 17α2 3α2 16α2 + n 1 2 3 G − 30 − 2 − 3 1 9α2 9α α 35α2 19α α 9α α 202α2 393α α + 1 1 2 2 + 1 3 + 2 3 3 +3λˆ2 + 1 t ǫ 400 − 40 − 8 30 2 − 3 160 225α α 7α α 99α α 11α α α2 + 2 t + 18α α 6λαˆ 6α2 + 1 b + 2 b +2α α t b b 32 3 t − t − t 160 32 3 b − 8 − 8 15α α 15α α 9α α 5α α 9α2 29α2 α2 40α2 + 1 τ + 2 τ t τ + b τ τ + n 1 + 2 + 3 , (B2) 16 16 − 8 8 − 8 G 90 2 9

1 1 α 9α 3α 3α Z =1+ 1 2 8α t + b + 3trTˆ + 3trBˆ + trLˆ αb 4π ǫ − 4 − 4 − 3 − 2 2 1 1 3α2 9α α 339α2 81α α 81α α + 1 + 1 2 + 2 +2α α + 18α α + 76α2 1 t 2 t 2 ǫ2 160 16 32 1 3 2 3 3 − 80 − 16 (4π) 9α2 27α α 243α α 45α α 81α2 11α α 18α α + t 1 b 2 b 54α α + t b + b 1 τ − 3 t 2 − 16 − 16 − 3 b 4 4 − 8 27α α 15α α 33α α 7α2 α2 3α2 16α2 2 τ 8α α + t τ + b τ + τ + n 1 2 3 − 8 − 3 τ 4 4 4 G − 6 − 2 − 3 1 29α2 27α α 35α2 31α α 9α α 202α2 91α α + 1 1 2 2 + 1 3 + 2 3 3 +3λˆ2 + 1 t ǫ − 400 − 40 − 8 30 2 − 3 160 99α α α2 237α α 225α α 11α α + 2 t +2α α t + 1 b + 2 b + 18α α 6λαˆ t b 6α2 32 3 t − 8 160 32 3 b − b − 8 − b 15α α 15α α 5α α 9α α 9α2 α2 α2 40α2 + 1 τ + 2 τ + t τ b τ τ + n 1 + 2 + 3 , (B3) 16 16 8 − 8 − 8 G − 90 2 9

1 1 9α 9α 3α Z =1+ 1 2 + τ + 3trTˆ + 3trBˆ + trLˆ ατ 4π ǫ − 4 − 4 2 2 2 2 1 1 387α1 81α1α2 339α2 321α1αt 81α2αt 45αt + + + 12α3αt + π 2 ǫ2 160 16 32 − 40 − 8 − 4 (4 ) 57α α 81α α 27α α 45α2 135α α 135α α 1 b 2 b 12α α + t b + b 1 τ 2 τ − 8 − 8 − 3 b 2 4 − 16 − 16 51α α 51α α 25α2 3α2 3α2 + t τ + b τ + τ + n 1 2 4 4 4 G − 2 − 2 37

1 51α2 27α α 35α2 17α α 45α α 27α2 + 1 + 1 2 2 +3λˆ2 + 1 t + 2 t + 10α α t ǫ 400 40 − 8 16 16 3 t − 8 5α α 45α α 3α α 27α2 537α α 165α α + 1 b + 2 b + 10α α + t b b + 1 τ + 2 τ 6λαˆ 16 16 3 b 4 − 8 160 32 − τ 27α α 27α α 3α2 11α2 α2 t τ b τ τ + n 1 + 2 . (B4) − 8 − 8 − 2 G 10 2

2 Appendix C: Beta functions for αQED and sin θW

This Appendix contains explicit results up to three-loop order for the QED coupling αQED and the weak mixing angle. We refrain from providing expressions for the renormalization constants but directly list the beta functions. They are obtained in a straightforward way from Eq. (2) and are given by

2 2 αQED 128nG αQED 500αQED 4αQED βα = 28 + + + QED 2 − 9 3 − 3 sin2 θ cos2 θ (4π) (4π) W W 52trTˆ 28trBˆ 208αQED 416αQED 320α3 12trLˆ + nG + + − 3 − 3 − 3 sin2 θ 27 cos2 θ 9 W W α2 137α2 318251α2 163α2 12α λˆ 8α λˆ + QED QED QED + QED + QED + QED 4 2 2 4 4 2 2 π 4 sin θW cos θW − 216 sin θW 72 cos θW sin θW cos θW (4 ) ˆ ˆ ˆ ˆ ˆ ˆ2 757αQEDtrT 2303αQEDtrT 200α3trT 583αQEDtrB 1433αQEDtrB 24λ 2 2 2 2 − − 4 sin θW − 36 cos θW − 3 − 4 sin θW − 36 cos θW ˆ ˆ ˆ ˆ2 ˆ ˆ 152α3trB 393αQEDtrL 183αQEDtrL 59trB ˆ 2 202trBtrL 2 2 + + 31(trB) + − 3 − 4 sin θW − 4 cos θW 2 3 53trLˆ2 85trTˆ2 244trTˆtrLˆ + + 19(trLˆ)2 + 16trTˆBˆ + + 104trTˆtrBˆ + + 73(trTˆ)2 2 2 3 2 2 2 32αQED 26146αQED 1580αQED 152αQEDα3 + nG + + 9 sin2 θ cos2 θ 27 sin4 θ − 81 cos4 θ 3 sin2 θ W W W W W 584α α 10000α2 1792α2 22880α2 3520α2 QED 3 + 3 + n2 QED QED 3 , (C1) − 81 cos2 θ 27 G − 27 sin4 θ − 729 cos4 θ − 81 W W W

2 2 d sin θW β 2 = µ sin θW dµ2 α sin2 θ 43 cos2 θ 20 sin2 θ 4 cos2 θ = QED W + W + n W W 4π 6 6 G 9 − 3 α α tan2 θ 259α cot2 θ 17 sin2 θ trTˆ + QED α + QED W + QED W W 2 QED 2 6 − 6 (4π) 3 cos2 θ trTˆ 5 sin2 θ trBˆ 3 cos2 θ trBˆ 5 sin2 θ trLˆ cos2 θ trLˆ + W W + W W + W 2 − 6 2 − 2 2 2α 95α tan2 θ 49α cot2 θ 44 sin2 θ α + n QED + QED W QED W + W 3 4 cos2 θ α G 3 27 − 3 9 − W 3 38

2 2 2 2 2 2 α 2279α 1403α 163α tan θW 667111α cot θW + QED QED + QED + QED + QED 3 192 sin2 θ 576 cos2 θ 576 cos2 θ 1728 sin2 θ (4π) W W W W 3α tan2 θ λˆ 3α cot2 θ λˆ + α λˆ + QED W QED W 3 sin2 θ λˆ2 + 3 cos2 θ λˆ2 QED 2 − 2 − W W 881α trTˆ 2827α tan2 θ trTˆ 729α cot2 θ trTˆ 29 sin2 θ α trTˆ QED QED W + QED W W 3 − 48 − 288 32 − 3 389α trBˆ 1267α tan2 θ trBˆ 729α cot2 θ trBˆ + 7 cos2 θ α trTˆ QED QED W + QED W W 3 − 48 − 288 32 17 sin2 θ α trBˆ 229α trLˆ 281α tan2 θ trLˆ W 3 + 7 cos2 θ α trBˆ QED QED W − 3 W 3 − 16 − 32 243α cot2 θ trLˆ 61 sin2 θ trBˆ2 57 cos2 θ trBˆ2 17 sin2 θ (trBˆ)2 + QED W + W W + W 32 16 − 16 8 45 cos2 θ (trBˆ)2 157 sin2 θ trBˆtrLˆ 15 cos2 θ trBˆtrLˆ 87 sin2 θ trLˆ2 W + W W + W − 8 12 − 4 16 19 cos2 θ trLˆ2 33 sin2 θ (trLˆ)2 5 cos2 θ (trLˆ)2 5 sin2 θ trTˆBˆ W + W W + W − 16 8 − 8 8 27 cos2 θ trTˆBˆ 113 sin2 θ trTˆ2 57 cos2 θ trTˆ2 59 sin2 θ trTˆtrBˆ W + W W + W − 8 16 − 16 4 45 cos2 θ trTˆtrBˆ 199 sin2 θ trTˆtrLˆ 15 cos2 θ trTˆtrLˆ W + W W − 4 12 − 4 2 2 2 2 2 2 101 sin θW (trTˆ) 45 cos θW (trTˆ) 127αQED 119αQED + + nG + 8 − 8 36 sin2 θ 108 cos2 θ W W 2 2 2 2 2 290αQED tan θW 6412αQED cot θW 2αQEDα3 137αQED tan θW α3 2 2 − 81 cos θW − 27 sin θW − 9 − 81 1375 sin2 θ α2 125 cos2 θ α2 13α cot2 θ α + W 3 W 3 − QED W 3 27 − 3 2 2 2 2 2 2 11α 55α 5225α tan θW 415α cot θW + n2 QED + QED QED + QED G − 9 sin2 θ 81 cos2 θ − 729 cos2 θ 27 sin2 θ W W W W 484 sin2 θ α2 44 cos2 θ α2 W 3 + W 3 . (C2) − 81 9

Appendix D: Comparison with Ref. [16]

In this Appendix we provide the explicit form of the expressions needed for the comparison with Ref. [16]. In this paper two-component Weyl spinors were used. To make contact with our convention based on four-component Dirac spinors we deﬁne χ ΨD = , (D1) ξ† where ξ and χ are left-handed Weyl spinors and ΨD denotes a Dirac spinor. Thus, the Lagrange density of the SM can be expressed in terms of 45 Weyl spinors:

χt ,ξt , χb ,ξb , χτ ,ξτ , χντ , χc ,ξc , χs,ξs, χµ,ξµ, χνµ , χu,ξu, χd,ξd, χe,ξe, χνe . (D2) 39

For simplicity we have suppressed the SU(3) color indices for all quark spinors. Of course, each quark spinor has to be understood as a triplet in color space. In this basis the Yukawa matrices become 45 45 dimensional. × In the notation of [16] the part of the Lagrange density describing the Yukawa couplings is given by

1 Y aijφaψ ψ + Y¯ aφaψ¯iψ¯j , (D3) −2 i j ij

where Y a and Y¯ a are the (complex conjugated) Yukawa matrices, φa are real scalar fields, ψi and ψ¯i are (Hermitian conjugated) spinor fields. There are four real scalar fields in the SM, which means that we have four Yukawa matrices. They are given by

0 y 11 0 0 0 0 0 3×3 t 3×3 3×3 3×3 3×1 3×1 3×1 ··· y 11 0 0 0 0 0 0  t 3×3 3×3 3×3 3×3 3×1 3×1 3×1 ··· 03×3 03×3 03×3 yb113×3 03×1 03×1 03×1  ··· 1  03×3 03×3 yb113×3 03×3 03×1 03×1 03×1  Y 1 =  ··· , √  01×3 01×3 01×3 01×3 0 yτ 0  2  ···  0 0 0 0 y 0 0   1×3 1×3 1×3 1×3 τ   ···  01×3 01×3 01×3 01×3 0 0 0   ...... ···.   ......    0 0 0 y 11 0 0 0  3×3 3×3 3×3 b 3×3 3×1 3×1 3×1 ··· 0 0 y 11 0 0 0 0  3×3 3×3 − t 3×3 3×3 3×1 3×1 3×1 ··· 03×3 yt113×3 03×3 03×3 03×1 03×1 03×1  − ··· 1 yb113×3 03×3 03×3 03×3 03×1 03×1 03×1  Y 2 =  ··· , √  01×3 01×3 01×3 01×3 0 0 0  2  ···  0 0 0 0 0 0 y   1×3 1×3 1×3 1×3 τ   ···  01×3 01×3 01×3 01×3 0 yτ 0   ...... ···.   ......     0 iy 11 0 0 0 0 0  3×3 − t 3×3 3×3 3×3 3×1 3×1 3×1 ··· iy 11 0 0 0 0 0 0 − t 3×3 3×3 3×3 3×3 3×1 3×1 3×1 ··· 03×3 03×3 03×3 iyb113×3 03×1 03×1 03×1  ··· 1  03×3 03×3 iyb113×3 03×3 03×1 03×1 03×1  Y 3 =  ··· , −√  01×3 01×3 01×3 01×3 0 iyτ 0  2  ···  0 0 0 0 iy 0 0   1×3 1×3 1×3 1×3 τ   ···  01×3 01×3 01×3 01×3 0 0 0   ...... ···.   ......      40

0 0 0 iy 11 0 0 0 3×3 3×3 3×3 b 3×3 3×1 3×1 3×1 ··· 0 0 iy 11 0 0 0 0  3×3 3×3 t 3×3 3×3 3×1 3×1 3×1 ··· 03×3 iyt113×3 03×3 03×3 03×1 03×1 03×1  ··· 1 iyb113×3 03×3 03×3 03×3 03×1 03×1 03×1  Y 4 =  ··· . (D4) −√  01×3 01×3 01×3 01×3 0 0 0  2  ···  0 0 0 0 0 0 iy   1×3 1×3 1×3 1×3 τ   ···  01×3 01×3 01×3 01×3 0 iyτ 0   ...... ···.   ......      In the above formulas, all matrix elements not explicitly given are zero. Y 1/Y 3 is the Yukawa matrix of the real/the imaginary part of the isospin down component of the SM Higgs doublet. Y 2/Y 4 is the Yukawa matrix of the real/the imaginary part of the isospin up component of the SM Higgs doublet. So we have

1 φ2 + iφ4 H = . (D5) √ φ1 + iφ3 2

The part of the Lagrange density describing the Higgs self-interaction reads in the notation of Ref. [16] 1 λ φaφbφcφd . (D6) −4! abcd

In the SM we have

λ =6 (4πλˆ), aaaa × λ = λ = λ =2 (4πλˆ) (a = b), aabb abab abba × 6 λabcd = 0 (otherwise) . (D7)

The expressions given above are generic for all three gauge groups. However, there are some expressions which depend on the specific gauge group one wants to consider. The remainder of this Section lists these expressions. For each of the three gauge groups, we give the expressions for the generators in the representations of the scalar fields, SA, and of the Weyl spinors, RA. We also give expressions for the invariants T (S),C(S), T (R),C(R),C(G),r, all of which are symbols used in Ref. [16]. They are defined as

A B AB A A Tr S S = δ T (S), SacScb = C(S)ab, A B AB Ak Aj j Tr R R = δ T (R), Ri Rk = C(R)i , ACD BCD AB AA f f = δ C(G), δ = r . (D8) f ABC are the structure constants of the respective gauge group. The indices A,B,C take values in the range 1,...,r, where r gives the dimension of the group. 41

1. U(1)

For the case of U(1) gauge group, we have

0 010 i 0 001 S1 =   , (D9) 2 1 0 00 −   0 100  −  1 1 1 2 2  2 1 1 1 11 1 1 1 R1 = Diag , , , , , , , , , , , , , 1, ,... , (D10) 6 6 6 −3 −3 −3 6 6 6 3 3 3 −2 −2 where the ellipsis is to be replaced twice by the ﬁrst 15 entries. The derivation of S1 is given below. The entries in R1 are the hypercharges Y of the respective spinors. They can be derived by using the relation Y = Q I , where Q corresponds to the electric charge and − 3 I3 is the weak isospin. Furthermore, we have

1 T (S)=1 , C(S)= 11 , 4 4×4 T (R)=10 , C(R)= R1 2 , C(G)=0 , r =1 . (D11)

Let us now we show how to derive Eq. (D9). We want to ﬁnd the U(1) representation transforming the four real scalar ﬁelds of the Higgs doublet of the SM (see Eq. (D5). The matrix S1 is the generator of this transformation,

′ φ1 φ1 φ1 ′ φ2 iωS1 φ2 1 2 φ2 ′ = e = 11+ iωS + (ω ) . (D12) φ3 φ3 O φ3  ′      φ  φ4 φ4  4           with ω being the transformation parameter. In a next step we take advantage of the fact that it is known how the SM Higgs doublet transforms in order to determine S1. As the SM Higgs doublet has hypercharge 1/2, we have

′2 ′4 2 4 1 φ + iφ ′ iω 1 11 iω 1 11 1 φ + iφ = H = e ( 2 )H = e ( 2 ) √ φ′1 + iφ′3 √ φ1 + iφ3 2 2 1 1 φ2 + iφ4 = 11+ iω 11 + (ω2) 2 O √ φ1 + iφ3 2 1 1 φ4 + iφ2 = H + ω − + (ω2) . (D13) 2 √2 φ3 + iφ1 O − With the help of the last equation one can determine the transformation of the SM Higgs ′ ω 2 1 doublet, like, e.g., φ1 = φ1 2 φ3 + (ω ). It is then straightforward to determine S by inserting the equations found− in thisO way in Eq. (D12). 42

2. SU(2)

For the SU(2) group, we have

00 0 1 0 10 0 0 01 0 − − i 0 0 1 0 i 1000 i 0 00 1 S1 =  −  , S2 =   , S3 =  −  , (D14) −2 010 0 −2 0 0 0 1 −2 100 0    −  −  100 0  0010   0 10 0              and σA 11 0 σA 11 0 0 0 0 1,1 3×3 3×3 1,2 3×3 3×3 3×1 3×1 3×1 ···  03×3 03×3 03×3 03×3 03×1 03×1 03×1  A A ··· σ2,1113×3 03×3 σ2,2113×3 03×3 03×1 03×1 03×1  ··· A  03×3 03×3 03×3 03×3 03×1 03×1 03×1  R =1/2  A A ··· . (D15)  01×3 01×3 01×3 01×3 σ 0 σ   2,2 2,1 ···  0 0 0 0 0 0 0   1×3 1×3 1×3 1×3   A A ···  01×3 01×3 01×3 01×3 σ1,2 0 σ1,1   ...... ···.   ......      In the last equation σA are the Pauli matrices. Not all of the matrix elements left out in RA vanish, but these elements play no role for the comparison of our results to [16]. The derivation of the generators SA proceeds in analogy to the derivation of S1 explained in 1 the former Subsection. One merely has to substitute the generator of U(1), 2 11, by the 1 A generators of SU(2), 2 σ , in Eq. (D13). We furthermore have 3 T (S)=1, C(S)= 11 , 4 4×4 3 T (R)=6, C(R)= Diag (1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1,...) , 4 C(G)=2, r =3 . (D16)

The ellipsis has to be replaced by the ﬁrst entries twice.

3. SU(3)

In the SM the matrices SA vanish for the group SU(3). The matrices RA are block- diagonal and read 1 RA = BlockDiag ΛA, (ΛA)T , ΛA, (ΛA)T , 0, 0, 0,... . (D17) 2 − − The ellipsis has to be replaced twice by the former entries, which contain the Gell-Mann matrices ΛA. Finally, we have

T (S)=0, C(S)=04×4, 43

4 T (R)=6, C(R)= BlockDiag (11 , 0 , 11 , 0 , 11 , 0 ) , 3 12×12 3×3 12×12 3×3 12×12 3×3 C(G)=3, r =8 . (D18)

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