arXiv:1210.6873v4 [hep-ph] 20 Mar 2013
Prepared for submission to JHEP
Anomalous dimensions of gauge fields and gauge coupling beta-functions in the Standard Model at three loops
A. V. Bednyakov,a A. F. Pikelnera and V. N. Velizhaninb a
Joint Institute for Nuclear Research, 141980 Dubna, Russia b Theoretical Physics Department, Petersburg Nuclear Physics Institute, Orlova Roscha, Gatchina, 188300 St. Petersburg, Russia
E-mail:
[email protected] Abstract: We present the results for three-loop gauge field anomalous dimensions in the SM calculated in the background field gauge within the unbroken phase of the model. The results are valid for the general background field gauge parameterized by three independent parameters. Both quantum and background fields are considered. The former are used to find three-loop anomalous dimensions for the gauge-fixing parameters, and the latter allow one to obtain the three-loop SM gauge beta-functions. Independence of beta-functions of gauge-fixing parameters serves as a validity check of our final results. Keywords: Renormalization Group, Standard Model ArXiv ePrint: 1210.6873
Contents 1 Introduction
1
2 The Standard Model in the unbroken phase. The background-field gauge
3
3 Details of calculations
5
4 Results and conclusions
8
A Renormalization constants
1
12
Introduction
In spite of the fact that the Standard Model has many unsatisfactory aspects Nature still does not allow us to find some solid evidence for the existence of a more fundamental theory with new particles and/or interactions. Due to the joint efforts of both experimentalists and theoreticians we are about to enter the only unexplored part of the SM and unveil the mechanism of electroweak symmetry breaking. According to the recent experimental results, there is strong evidence for the existence of the Higgs boson, the last missing ingredient of the SM spectrum [1, 2]. The mass of the higgs seems to be located at the boundary of the so-called stability and instability regions [3–5] in the SM phase diagram (see Refs. [6–8] for recent studies). This fact implies that the SM can be potentially valid up to a very high scale (e.g., Plank scale). In this situation, it is important to know how the running SM parameters evolve with energy scale. The analysis of high energy behavior is usually divided into two parts. The first one is the determination of running MS-parameters from some (pseudo)observables. This procedure is usually referred to as “matching”. The second one utilizes renormalization group equations (RGEs) to find the corresponding values at some “New Physics” scale. In order to carry out such an analysis consistently one usually use (L − 1)-loop matching to find boundary conditions for L-loop RGEs (see, e.g., [9]). It is worth pointing that the advantage of the minimal-subtraction prescription lies in the fact that one needs to know only the ultraviolet (UV) divergent part of all the required diagrams. The latter has a simple polynomial structure in mass and momenta (once subdivergences are subtracted). Due to this, MS beta functions and anomalous dimensions can be relatively easily extracted from Green functions by solving a single scale problem with the help of the so-called infrared rearrangements (IRR) [10]. One- and two-loop results for SM beta functions have been known for quite a long time [11–21] and are summarized in [22]. Until recently, three-loop corrections were known only partially [23–28].
–1–
Having a well tested method for calculation of three-loop renormalization constants [29– 31] and an experience in the calculations in the Standard Model and its minimal supersymmetric extension [32–34] we are planning to perform the calculation of all renormalization group coefficients in the third order of perturbation theory extending the results of Refs. [19, 35, 36] to one more loop. In this paper, we present our first step in this direction: the results for three-loop anomalous dimensions of the SM gauge fields. Since we are only interested in UV-divergences for the fields and dimensionless parameters, we do not consider the effects related to spontaneous breaking of electroweak symmetry and, as a consequence, can neglect all dimensionful parameters of the model. Moreover, we made use of the background-field gauge (BFG) (see, e.g., Refs. [37, 38]) to carry out our calculation. In this gauge, due to the simple QEDlike Ward identities involving background fields, one can easily obtain expressions for the beta-functions by considering the two-point functions with external background particles. During the work on this project a few papers on the same topic appeared [39, 40] (gauge couplings) and [41] (Top Yukawa and higgs self-interactions). Since the authors of [39, 40] carried out a similar calculation, let us mention that our setup differs from that used in Ref. [39, 40] in several aspects. Firstly, for the diagram generation we solely rely on FeynArts [42]. Since the diagrams are evaluated with the help of the MINCER package [43], a mapping to the MINCER notation for momenta is required. This problem was solved by hand with the help of the DIANA [44] topology files which were prepared during our previous calculations [29]. Based on these files a simple script was written which allows one to perform the mapping between the FeynArts and MINCER notation1 . Secondly, we do not consider the unbroken SM in a general Lorentz gauge, in which case we are forced to take into account vertex renormalization, but choose to work within the unbroken SM in a general background-field gauge. We keep the full dependence of the diagrams on the electroweak gauge-fixing parameters and take into account corresponding renormalization. Absence of these auxiliary parameters in the final expressions for betafunctions gives us an independent confirmation of the correctness of our calculation. It is worth mentioning that in Refs. [39, 40] the SM in BFG was also considered. However, the corresponding calculation was carried out in the spontaneously broken phase and the model file distributed with FeynArts package was used. Since a consistent renormalization of the electroweak gauge-fixing parameters in the spontaneously broken phase requires a severe modification of corresponding part of the model file (see, e.g., Refs. [45–47]), the Landau gauge was chosen in [40] to avoid these kind of problems. And lastly, since the unbroken SM in BFG is not implemented as a FeynArts model file, we are forced to use a package like FeynRules [48] or LanHEP [49]. Due to the fact that the authors are more accustomed to the latter, LanHep was chosen to generate the required Feynman rules from the Lagrangian2 . 1
During the preparation of the final version of the paper a routine was written that automatically maps the FeynArts topologies onto that of MINCER. 2 The authors of Refs. [39, 40] utilize FeynRules to obtain a model file for the unbroken SM.
–2–
The paper is organized as follows. In Section 2 we introduce our notation and present a brief description of the unbroken SM quantized in the background-field gauge. Section 3 describes the details of our calculation strategy. Finally, the results and conclusions can be found in Section 4. Appendix contains all the expressions for the considered renormalization constants.
2
The Standard Model in the unbroken phase. The background-field gauge
Let us briefly review the Lagrangian of the SM in the background-field gauge. We closely follow [50] albeit the fact that we introduce background fields only for gauge bosons. Moreover, as it was mentioned in Introduction, we neglect all the dimensionful couplings (i.e., mass parameters). In our calculation we use the Lagrangian of the form L = LG + LH + LF + LGF + LFP .
(2.1)
Here LG is the Yang-Mills part 1 i 1 1 i Wµν − Bµν Bµν , LG = − Gaµν Gaµν − Wµν 4 4 4 Gaµν = ∂µ Gaν − ∂ν Gaµ + gs f abc Gbµ Gcν , i Wµν
=
∂µ Wνi
−
∂ν Wµi
+ g2 ǫ
ijk
(2.2) (2.3)
Wµj Wνk ,
(2.4)
Bµν = ∂µ Bν − ∂ν Bµ ,
(2.5)
˜ aµ + G ˆ aµ (a = 1, . . . , 8), Wµi = W ˜ µi + W ˆ µi , (i = 1, 2, 3), and Bµ = B ˜µ + B ˆµ where Gaµ = G ˜ W ˜ , B) ˜ we denote quantum are gauge fields for SU(3), SU(2) and U(1) groups. By V˜ = (G, ˆ ˆ ˆ ˆ fields, and V = (G, W , B) is used for their background counterpart. The corresponding gauge couplings are gs , g2 , and g1 . The group structure constants enter into the commutation relations i h i j τ , τ = iǫijk τ k , (2.6) T a , T b = if abc T c , with T a = λa /2 and τ i = σ i /2 being color and weak isospin generators. The covariant derivative acting on a field which is charged under all the gauge groups looks like YW Bµ . (2.7) Dµ = ∂µ − igs T a Gaµ − ig2 τ i Wµi + ig1 2 If a field is not charged under either group, the corresponding term is omitted. With the help of the covariant derivative one can write the following Higgs and fermionic parts of the Lagrangian: 2 LH = (Dµ Φ)† (Dµ Φ) − λ Φ† Φ , (2.8) X ¯R ˆ R ¯R ˆ R ¯L ˆ L ¯L ˆ L ˆ R LF = iQ uR i DQi + iLi DLi + i¯ g Dug + idg Ddg + ilg Dlg i=1,2,3
−
X
c R Yuij (QL i Φ )uj
+
R Ydij (QL i Φ)dj
i,j=1,2,3
–3–
+
R Ylij (LL i Φ)lj
+ h.c. ,
(2.9)
where indices i, j = 1, 2, 3 count different fermion families, λ and Yu,d,l are the higgs quartic L and Yukawa matrices3 , respectively. The left-handed quarks QL g = (ug , dg ) and leptons L R R LL g = (νg , lg ) form the SU(2) doublets while the right-handed quarks (ug , dg ) and charged leptons lgR are the singlets with respect to SU(2). The Higgs doublet Φ with YW = 1 has the following decomposition in terms of the component fields: ! ! √1 (h − iχ) φ+ (x) c 2 † 2 . (2.10) Φ = √1 , Φ = iσ Φ = (h + iχ) −φ− 2 Here a charge-conjugated Higgs doublet is introduced Φc with YW = −1. The gauge-fixing terms are introduced only for quantum fields 1 a a 1 1 2 LGF = − GG GG − GiW GiW − G , 2ξG 2ξW 2ξB B with
(2.11)
˜ aµ + gs f abc G ˆ bµ G ˜ cµ , GaG = ∂µ G ˜ µi + g2 ǫijk W ˆ µj W ˜ µk , GiW = ∂µ W ˜µ . GB = ∂µ B
(2.12)
The ordinary derivatives are replaced by covariant ones containing the background fields. Due to this, the invariance of the effective action under background gauge transformations is not touched by introduction of (2.11). The Fadeev-Popov part of the Lagrangian is given by δGα (2.13) LFP = −¯ cα β cβ δθ where α, β = (G, W, B), and δGα /δθ β is the variation of gauge-fixing functions (2.12) under the following infinitesimal quantum gauge transformations ˜ a = (Dµ θG )a = ∂µ θ a + gs f abc Gb θ c , δG µ G µ G i i i ijk k ˜ δWµ = (Dµ θW ) = ∂µ θW + g2 ǫ Wµj θW , ˜µ = ∂µ θB . δB
(2.14)
It should be stressed that covariant derivatives in (2.14) involve the sum of quantum and background gauge fields V = V˜ + Vˆ . The corresponding background transformations are obtained from (2.14) by the replacement V → Vˆ . The Feynman rules for the model described by the Lagrangian (2.1) were generated with the help of LanHEP 4 [49]. It is worth mentioning here that our problem does not require the introduction of U(1) ˆ fields. This is due to the fact that the latter has the same ghosts c¯B , cB and background B ˜ and the former decouples from other particles. interactions as its quantum counterpart B Nevertheless, we keep them in our LanHEP model file to allow for possible generalizations to non-linear gauge-fixing as in Ref. [50]. 3
In the actual calculation the diagonal Yukawa matrices were used. However, the result can be generalized with the help of additional tricks (see Sec.3 and Ref. [40]). 4 LanHEP 3.1.5, which was used by the authors, produces a wrong sign for the combination f abc f dec during export to the FeynArts model files. A new version with a fix is scheduled for November 2012.
–4–
3
Details of calculations
Due to the gauge invariance of the effective action for the background fields, QED-like Ward identities can be derived. The latter can be used to prove the following simple relations: −1/2 , Vi
Zgi = Z ˆ
i = 1, 2, 3
(3.1)
ˆ µ, W ˆ µ, G ˆµ) with ZVˆi and Zgi being renormalization constants for background fields Vˆiµ = (B and SM gauge couplings gi = (g1 , g2 , gs ), respectively. Since we keep the full dependence on the gauge-fixing parameters ξi during the whole calculation, we also need to know how ξi = (ξB , ξW , ξG ) are renormalized. Again, due to the Ward identities, the longitudinal part of the quantum gauge field propagators does not receive any loop corrections. As a consequence, the following identities hold: Zξi = ZV˜i .
(3.2)
Here Zξi stands for the renormalization constants for the gauge-fixing parameters. The quantum gauge fields V˜i are renormalized in the MS-scheme with the help of ZV˜i . It is clear from (3.1) and (3.2) that to carry out the calculation, one needs to consider gauge boson self-energies for both quantum V˜ and background Vˆ fields. For calculation of the renormalization constants, following [26] (see also [10, 14, 51]), we use the multiplicative renormalizability of the corresponding Green functions. The renormalization constants ZV relate the dimensionally regularized one-particle-irreducible two-point functions ΓV,Bare with the renormalized one ΓV,Ren as: 2 Q 1 2 ΓV,Ren , a , a Q , a , ǫ , (3.3) = lim Z Γ i i i,Bare V V,Bare ǫ→0 µ2 ǫ where ai,Bare are the bare parameters of the model. For convenience, we introduce the following notation, which is closely related to that used in Ref. [40], Yd2 Yl2 5 g12 g22 gs2 Yu2 λ ai = , , , , , , , ξG , ξW , ξG , (3.4) 3 16π 2 16π 2 16π 2 16π 2 16π 2 16π 2 16π 2
so we treat the gauge-fixing parameters along the same lines as couplings. Moreover, in the renormalization group analysis of the SM one usually employs the SU(5) normalization of the U(1) gauge coupling which leads to an additional factor 5/3 in (3.4). The bare parameters are related to the renormalized ones in the MS-scheme by the following formula: ∞ X (n) 1 −2ρk ǫ (3.5) ck n , ak,Bare µ = Zak ak (µ) = ak + ǫ n=1
where ρk = 1/2 for the gauge (g1 , g2 , gs ) and Yukawa constants (Yu , Yd , Yl ), ρk = 1 for the scalar quartic coupling constant λ, and ρk = 0 for the gauge fixing parameters. In order to extract a three-loop contribution to ZV from the corresponding self-energies, it is sufficient to know the two-loop renormalization constants for the gauge couplings and the one-loop results for the Yukawa couplings. This is due to the fact that the Yukawa vertices appear
–5–
for the first time only in the two-loop self-energies and the higgs self-coupling enters into the result only at the third level of perturbation theory. The four-dimensional beta-functions, denoted by βi , are defined via dai (µ, ǫ) . (3.6) βi (ak ) = d ln µ2 ǫ=0
Here, again, ai stands for both the gauge couplings and the gauge-fixing. Given the fact that the bare parameters do not depend on the renormalization scale the expressions for βi can be obtained [19] by differentiation of (3.5) with respect to ln µ2 :
− ρk ǫ ak +
∞ X
n=1
(n) 1 ck n ǫ
∞ X (n) X ∂ck 1 . = −ρk ǫak + βk + (βl − ρl al ǫ) ∂al ǫn n=1
(3.7)
l
Taking in account only the leading order of the expansion in ǫ: (1)
βk =
X
ρ l al
l
∂ck (1) − ρk ck . ∂al
(3.8)
In MS-like schemes the renormalization constants for the Green functions may be expanded as ∞ (k) X ZΓ ZΓ = 1 + . (3.9) ǫk k=1
ln µ2
Differentiating (3.9) with respect to we simply get all-order expression for anomalous dimensions: X ∂ZΓ −1 ∂ ln ZΓ = − (3.10) βj − ρj aj ǫ ZΓ . γΓ ≡ −µ2 ∂µ2 ∂aj j
It turns out that the above expression is finite as ǫ → 0 so (1)
γΓ =
X
aj ρj
j
∂ZΓ . ∂aj
(3.11)
The advantage of (3.7) and (3.10) comes from the fact that it provides us with additional confirmation of the correctness of the final result since beta functions and anomalous di(n) mensions extracted directly from (3.7) and (3.10) are finite for ǫ → 0 only if ck satisfy the so-called pole equations [52], e.g., # " X ∂c(n) X ∂ (n+1) = βl k . − ρk ck (3.12) ρl al ∂al ∂al l
l
In order to calculate the bare two-point functions for the quantum and background fields, we generate the corresponding diagrams with the help of the FeynArts package [42]. It is worth pointing that we use the Classes level of diagram generation which allows us to significantly reduce the number of generated diagrams since we do not distinguish fermion generations. The complexity of the problem can be deduced from Table 1 that shows how
–6–
the number of the FeynArts generated diagrams increases with the loop level. Clearly, the presented numbers are an order of magnitude less than those given in Table I of Ref. [40], which somehow demonstrate the advantage of our approach. The number of the SM fermion generations is introduced via counting fermion traces present in the generated expression for a diagram and multiplying it by nG . We separately count fermion traces involving the Yukawa interaction vertices and multiply them not by nG but by nY . This allows us to use the following substitution rules (c.f., [40]) to generalize the obtained expression to the case of the general Yukawa matrices nY au , ad , al → Yu , Yd , Yl , nY a2u , a2d , a2l → Yuu , Ydd , Yll , n2Y a2u , a2d , a2l → Yu2 , Yd2 , Yl2 , n2Y au ad , ad al , au al → Yu Yd , Yd Yl , Yu Yl , nY au ad → Yud
where Yu =
tr Yu Yu† , 16π 2
Yd =
(3.13)
tr Yd Yd† , 16π 2
Yl =
tr Yl Yl† , 16π 2
(3.14)
and Yuu =
tr Yu Yu† Yu Yu† , (16π 2 )2
Ydd =
Yud =
tr Yu Yu† Yd Yd† , (16π 2 )2
Yll =
tr Yd Yd† Yd Yd† , (16π 2 )2 tr Yl Yl† Yl Yl† . (16π 2 )2
(3.15)
A comment is in order about the last substitution in (3.13). It turns out that Yud is the only combination of up- and down-type Yukawa matrices, which can appear in the result for the three-loop gauge-boson self-energy within the SM. This can be traced to the following facts: 1) in the unbroken SM all the particles are massless so that chirality is conserved during fermion propagation; 2) only the Yukawa interactions flip the chirality of the incoming fermions; 3) there is no right-handed flavour changing current coupled to a SM gauge field. As a consequence, combinations like tr Yu Yd† Yu Yd† (16π 2 )2
and
tr Yu Yd† Yd Yu† , (16π 2 )2
(3.16)
which require at least two chirality-conserving transitions between right-handed up- and down-type quarks, do not show up in the result. This type of counting is performed at the generation stage. A simple script converts the output of FeynArts to DIANA-like [44] notation and identifies MINCER topologies. This allows us to use the FORM [53] package COLOR [54] to do the SU(3) color algebra and MINCER [43] to obtain the ǫ-expansion of diagrams. It is worth pointing that the expressions for all SM gauge couplings exhibit explicit dependence on number of colors Nc which stems from the fact that we have to sum over color when there is a (sub)loop with external color singlets coupled to quarks.
–7–
Broken W + /W − Z A ZA G Total
1 10 9 7 7 4 37
2 339 281 218 236 67 1141
3 21942 19041 14426 16120 3287 74816
Unbroken ˆi W ˜i W ˆ B ˜ B, ˆ G ˜ G Total
1 11 11 6 4 4 36
2 389 371 214 73 66 1113
3 36647 36103 20144 4183 4060 101137
Table 1. Number of self-energy diagrams with external gauge fields, generated by FeynArts in the broken and unbroken SM, at one, two, and three loops.
During our calculation we made use of naive anticommuting prescription for dealing with γ5 (see, e.g., a nice review [60]). In this case, however, closed fermion loops with odd number of γ5 (“odd traces”) are not treated properly. In D = 4 such traces inevitably lead to the appearance of four-dimensional antisymmetric tensors ǫµνρσ that in the final result for a diagram should be contracted either between themselves or with external Lorentz indices and/or momenta. Since we are only interested in two-point functions the only non-zero combination that could potentially appear after loop integration is ǫµαρσ ǫνβρσ qα qβ which originates at three loops from two odd traces. In this expression q corresponds to external momentum, µ, ν denote external Lorentz indices, and ρ, σ are dummy indices representing the contractions due to internal vector boson propagators. A simple counting shows that both closed fermion lines are one-loop triangle (sub)graphs contributing to Adler-BellJackiw gauge anomalies [55–57]. Having in mind the cancellation of such anomalies within the SM [58, 59], in our calculation we can safely put all the Dirac traces involving odd number of γ5 to zero (see also the discussion in Ref. [40]). It is worth mentioning that the correct results for three-loop contribution to Yukawa coupling beta-functions [41] can not be obtained without special treatment of such traces.
4
Results and conclusions
Here we present the results of our calculations in the form of the SM gauge beta-functions and anomalous dimension of the gauge-fixing parameters. From (3.1) and (3.2) it is clear that anomalous dimensions of the background fields are connected with the corresponding gauge coupling beta-functions γBˆ = −β1 /a1 ,
γW ˆ = −β2 /a2 ,
γGˆ = −βs /as
(4.1)
γW ˜ = βξW /ξW ,
γG˜ = βξG /ξG .
(4.2)
and for the quantum fields we have γB˜ = βξB /ξB ,
The corresponding renormalization constants can be found in the Appendix. ˆ ≡ aλ ): At the end of the day, we have the following expressions for the beta-functions (λ 11 Nc 3 1 2 β1 = a1 nG + + 45 5 10
–8–
137a1 Nc 81a1 a2 Nc 9a2 11as CF Nc + nG + + + + 900 100 20 20 15 9a1 9 a2 Nc Yd 17Nc Yu 3Yl + + − − − 50 10 6 30 2 2 2 a1 a2 Nc 27a1 a2 1697a1 Nc 981a1 − − − + a21 nG − 18000 2000 1200 400 2 2 137 a Nc 27a2 1 1463 2 − a1 as CF Nc + 2 + − a2 as CF Nc + as CA CF Nc 900 45 10 20 540 11a22 Nc2 16577a21 Nc2 2387a21 Nc 891a21 11 − − − − a2s CF2 Nc + n2G − 30 486000 9000 2000 720 ˆ 11a22 Nc 11 a22 242 2 489a21 783a1 a2 27a1 λ − − − as CF TF Nc + + + 72 80 135 8000 800 50 ˆ 1267a1 Nc Yd 2827a1 Nc Yu 2529a1 Yl 3401 a22 9a2 λ − − + + − 2400 2400 800 320 10 29 437a2 Nc Yd 157a2 Nc Yu 1629a2 Yl 17 − − − as CF Nc Yd − as CF Nc Yu − 160 32 160 20 20 2 2 2 2 2 ˆ 17Nc Yd 59 101 Nc Yu 157Nc Yd Yl 61Nc Ydd 9λ + + Nc2 Yd Yu + + + − 5 120 60 120 60 80 2 199Nc Yl Yu Nc Yud 113 Nc Yuu 99Yl 261 Yll + + + + + , (4.3) 60 8 80 40 80 a21
43 Nc 1 + β2 = − nG 3 3 6 3 a 49a2 Nc 49 a2 a N 1 1 c 2 + + + + as CF Nc + a2 n G 60 20 12 12 3 a1 259a2 Nc Yd Nc Yu Yl + − − − − 10 6 2 2 2 2 2 13a1 a2 Nc 39a1 a2 1 287a1 Nc 91a1 + a22 nG − − + + − a1 as C F Nc 3600 400 240 80 60 13 133 2 1 2 2 1603a22 Nc 1603a22 + + a2 as CF Nc + a CA CF Nc − as CF Nc + 27 27 4 36 s 2 2 2 2 2 2 2 2 77a1 Nc 33 a1 415a2 Nc 415 a2 Nc 415a22 121a1 Nc + n2G − − − − − − 32400 1800 400 432 216 432 2 ˆ 22 2 163 a1 561a1 a2 3a1 λ 533a1 Nc Yd 593 a1 Nc Yu as CF TF Nc + + + − − − 9 1600 160 10 480 480 2 ˆ 51a1 Yl 667111a2 3a2 λ 243a2 Nc Yd 243 a2 Nc Yu 243a2 Yl − − + − − − 32 1728 2 32 32 32 2Y 2 2 2 5N 7 5 5N 7 c d c Yu ˆ2 + + Nc2 Yd Yu + − as CF Nc Yd − as CF Nc Yu − 3 λ 4 4 8 4 8 5Nc Yd Yl 19Nc Ydd 5 Nc Yl Yu 9Nc Yud 19Nc Yuu + + + + + 4 16 4 8 16 2 5Yl 19Yll + + , (4.4) 8 16 a22
–9–
8TF nG 11 CA βs = a2s − 3 3 40as CA TF 11a1 TF 2 + as n G + 3 a2 TF + + 8 as CF TF 15 3 2 34as CA a1 a2 TF 13a21 TF 2 − 4 TF Yd − 4TF Yu + as nG − − − 3 60 20 2 22 11 241a2 TF + a1 as CA TF − a1 as CF TF + 15 15 12 410 2 2830 2 2 2 2 a C TF + a CA CF TF − 4as CF TF + 6a2 as CA TF − 3a2 as CF TF + 27 s A 9 s 1331a21 TF Nc 121a21 TF 11 2 11a22 TF 632 2 2 + nG − − − a2 TF Nc − − a CA TF2 8100 300 12 12 27 s 176 2 89a1 TF Yd 101 a1 TF Yu 93a2 TF Yd 93a2 TF Yu 2 − a CF TF − − − − 9 s 20 20 4 4 2857 2 3 a C − 24as CA TF Yd − 24 as CA TF Yu − 6as CF TF Yd − 6as CF TF Yu − 54 s A + 7TF Nc Yd2 + 14TF Nc Yd Yu + 7 TF Nc Yu2 + 7TF Yd Yl + 9 TF Ydd + 7TF Yl Yu (4.5) − 6TF Yud + 9TF Yuu , βξB
3a1 11a1 Nc a1 + nG − − = ξB − 10 5 45 2 9a 9a1 a2 a1 Nc Yd 3a1 Yl 17a1 Nc Yu + ξB − 1 − + + + 50 10 6 2 30 2 2 81a1 9a1 a2 137a1 Nc a1 a2 Nc 11 + nG − − − − − a1 as CF Nc 100 20 900 20 15 ˆ 9a1 a2 λ ˆ 9a1 λ ˆ2 783a21 a2 3401a1 a22 27a21 λ 489a31 − − − − + + ξB − 8000 800 320 50 10 5 3 2 2 3 2 981a1 27a1 a2 27a1 a2 1697a1 Nc a1 a2 Nc 1 + nG + − + + − a1 a22 Nc 2000 400 10 18000 1200 45 1 1463 11 137 2 2 2 2 a as CF Nc + a1 a2 as CF Nc − a1 as CA CF Nc + a1 as CF Nc + 900 1 20 540 30 3 2 3 891a1 11a1 a2 2387a1 Nc 11 242 + n2G + + + a1 a22 Nc + a1 a2s CF TF Nc 2000 80 9000 72 135 16577a31 Nc2 11 1267a21 Nc Yd 437 2 2 + + a1 a2 Nc + + a1 a2 Nc Y d 486000 720 2400 160 61a1 Nc Ydd 2529a21 Yl 1629a1 a2 Yl 17 17 a1 Nc2 Yd2 − + + + a1 as CF Nc Yd − 20 120 80 800 160 99a1 Yl2 261a1 Yll 2827a21 Nc Yu 157 157 a1 Nc Yd Yl − − + + a1 a2 Nc Yu − 60 40 80 2400 32 29 59 199 101 + a1 as CF Nc Yu − a1 Nc2 Yd Yu − a1 Nc Yl Yu − a1 Nc2 Yu2 20 60 60 120
– 10 –
a1 Nc Yud 113a1 Nc Yuu − − 8 80
βξW
,
(4.6)
a2 a2 Nc 25a2 − a2 ξ W + n G − − = ξW 6 3 3 2 2 3a1 a2 113a2 11a2 ξW 3a1 a2 2 2 + ξW − + − − a2 ξ W + n G − 10 4 2 20 2 2 13a2 a1 a2 Nc 13a2 Nc a2 Nc Yd a2 Yl a2 Nc Yu − − − − a2 as CF Nc + + + 4 60 4 2 2 2 ˆ ˆ 3a2 λ 3a1 a2 λ 163a21 a2 33a1 a22 143537a32 ˆ2 − + − − 2 + 3a2 λ + ξW − 1600 32 576 10 2 2 3 33a32 ξW 7a32 ξW 315a32 ξW 77a21 a2 Nc 33a21 a2 185a32 2 − − − + nG + + 8 4 4 400 144 1800 2 2 3 2 3 121a1 a2 Nc 185a2 Nc 533 185a2 Nc 22 + a2 a2s CF TF Nc + + a1 a2 Nc Yd + + 72 9 32400 144 480 79 7 5 19a2 Nc Ydd 51a1 a2 Yl + a22 Nc Yd + a2 as CF Nc Yd − a2 Nc2 Yd2 − + 32 4 8 16 32 5a2 Yl2 19a2 Yll 593 79 79a22 Yl 5 − a2 Nc Y d Y l − − + a1 a2 Nc Yu + a22 Nc Yu + 32 4 8 16 480 32 5 5 5 9a 7 2 Nc Yud + a2 as CF Nc Yu − a2 Nc2 Yd Yu − a2 Nc Yl Yu − a2 Nc2 Yu2 − 4 4 4 8 8 3 ζ(3) 9 3a 3 19a2 Nc Yuu 2 − a1 a22 ζ(3) + 2 − 6a32 ξW ζ(3) − a32 ξW ζ(3) − 16 10 2 2 91a21 a2 6a1 a22 7025a32 287a21 a2 Nc 2 + nG + − + 2a32 ξW + + a1 a22 Nc 400 5 144 3600 15 3 1 133 7025a2 Nc + a1 a2 as CF Nc + 8a22 as CF Nc − a2 a2s CA CF Nc − 144 60 36 1 9 1 + a2 a2s CF2 Nc + 2a32 ξW Nc − a1 a22 ζ(3) + 9a32 ζ(3) − a1 a22 Nc ζ(3) 2 5 5
+ 9a32 Nc ζ(3) − 12a22 as CF Nc ζ(3)
βξG
,
13as CA as CA ξG 8as TF nG = ξG − − 6 2 3 2 2 59as CA 11 2 2 1 2 2 + ξG − as CA ξG − a2s CA ξG + 4as TF Yd + 4as TF Yu 8 8 4 11 + − a1 as TF − 3a2 as TF − 10a2s CA TF − 8a2s CF TF nG 15 3 9965a3s CA 167 3 3 33 3 3 2 7 3 3 3 121 2 2 − as C A ξ G − as C A ξ G − as C A ξ G + n G a as TF + ξG 288 32 32 32 300 1 11 2 304 3 176 3 1331a21 as TF Nc 11 2 2 2 + a2 as TF + a CA TF + a CF TF + + a2 as TF Nc 12 9 s 9 s 8100 12
– 11 –
(4.7)
93 25 89 a1 as TF Yd + a2 as TF Yd + a2s CA TF Yd + 6a2s CF TF Yd − 7as TF Nc Yd2 20 4 2 101 93 25 − 9as TF Ydd − 7as TF Yd Yl + a1 as TF Yu + a2 as TF Yu + a2s CA TF Yu 20 4 2 + 6a2s CF TF Yu − 14as TF Nc Yd Yu − 7as TF Yl Yu − 7as TF Nc Yu2 + 6as TF Yud 9 3 3 3 3 3 3 3 3 2 13 2 − 9as TF Yuu − as CA ζ(3) − as CA ξG ζ(3) − as CA ξG ζ(3) + nG a as TF 16 4 16 60 1 241 2 319 87 911 3 2 1 a as TF + a1 a2s CA TF + a2 a2s CA TF − a C TF + a1 a2 as TF − 20 12 2 120 8 9 s A 11 5 2 + a1 a2s CF TF + 3a2 a2s CF TF − a3s CA CF TF + 4a3s CF2 TF + 4a3s CA ξG TF 15 9 22 2 − a1 a2s CA TF ζ(3) − 18a2 a2s CA TF ζ(3) + 36a3s CA TF ζ(3) 5 +
− 48a3s CA CF TF ζ(3)
.
(4.8)
With the help of substitutions CA = Nc = 3, CF = 4/3, TF = 1/2, Yu = tr Tˆ, ˆ Yl = tr L, ˆ Ydd = tr(B ˆ 2 ), Yuu = tr(Tˆ2 ), Yll = tr(L ˆ 2 ), and Yud = tr TˆB ˆ it is Yd = tr B, possible to prove that the expressions presented above coincide with the results for the gauge beta functions obtained in Ref. [39]. As a consequence, one can be sure that the three-loop renormalization group equations obtained for the first time in Ref. [39] are correct and confirmed by an independent calculation. It is also worth mentioning that the obtained results can be used not only for the analysis of vacuum stability constraints within the SM (as in Refs. [6–8]) but also, e.g., for very precise matching of the SM with its supersymmetric extension since the corresponding three-loop renormalization group functions are already known from the literature [61–63]. Moreover, the leading two-loop decoupling corrections for the strongest SM couplings are also calculated within the MSSM in Refs. [32, 33, 64, 65]. Acknowledgments The authors would like to thank M. Kalmykov for drawing our attention to the problem and A. Semenov for his help with the LanHEP package. This work is partially supported by RFBR grants 11-02-01177-a, 12-02-00412-a, RSGSS-4801.2012.2
A
Renormalization constants
Here we present the results for the renormalization constants from which the anomalous dimensions and beta-functions were extracted. It should be pointed out that the coefficients of the ǫ-expansion satisfy the pole equations (3.12). The corresponding expressions together with the results for beta-functions can be found online5 in the form of Mathematica files.
Zα1 5
3 1 1 11Nc + = 1 + a1 nG + ǫ 45 5 10
As ancillary files of the arXiv version of the paper
– 12 –
1 2 121a1 Nc2 22a1 Nc 9a1 a1 11a1 Nc 3a1 + a1 2 n G + + + + nG + ǫ 2025 75 25 225 25 100 9a2 9a1 137a1 Nc 81a1 a2 Nc 9a2 11as CF Nc 1 + + + + + + + nG ǫ 1800 200 40 40 30 100 20 Nc Yd 17Nc Yu 3Yl − − − 12 60 4 2 1 1331a1 Nc3 121a21 Nc2 33a21 Nc 27a21 + a1 3 n3G + + + ǫ 91125 1125 125 125 2 2 2 2 2 121a1 Nc 11a1 Nc 27a1 9a21 a21 11a1 Nc 2 + nG + + + + nG + 6750 125 250 1500 500 1000 2 2 9a1 a2 Nc 117a1 a2 11 3731a1 Nc 441a1 1 + + + + a1 as CF Nc + 2 nG ǫ 54000 2000 40 200 150 11 187a1 Nc2 Yu a1 Nc Yd 11a1 Nc Yl 17a1 Nc Yu 9a1 Yl − a1 Nc2 Yd − − − − − 270 1350 10 30 50 10 2 2 2 2 2 1519a1 Nc 10549a1 Nc 7a Nc 39a2 121 2 − − a CA CF Nc + n2G + − 2 720 80 270 s 243000 4500 2 567a1 11 7a1 a2 Nc 27a1 a2 121 11 + + a1 a2 Nc2 + + + a1 as CF Nc2 + a1 as CF Nc 1000 900 50 100 675 25 a22 44 2 9a1 a2 7a1 Nc Yd 21a21 a22 Nc2 a22 Nc + + + a CF TF Nc + + − + 360 36 40 135 s 1000 100 720 2 17a1 Nc Yu 33a1 Yl 43a2 a2 Nc Yd 17a2 Nc Yu 9a2 Yl + + − + + + 720 80 40 16 80 16 Nc2 Yd2 11 2 17 17Nc2 Yu2 1 − Nc Y d Y u − + as CF Nc Yd + as CF Nc Yu − 6 30 36 90 180 5Nc Yd Yl Nc Ydd 31Nc Yl Yu 11Nc Yud 17Nc Yuu Yl2 3Yll − − + − − − − 18 24 90 60 120 4 8 2 2 1 a1 a2 Nc 9a1 a2 137a1 as CF Nc 1697a1 Nc 327a1 + nG − − − − − ǫ 54000 2000 3600 400 2700 2 2 2 1 1463as CA CF Nc 11 2 2 a Nc 9a2 + − a2 as CF Nc + − as C F Nc + 2 135 10 60 1620 90 2 2 2 2 2 16577a1 Nc 2387a1 Nc 297a1 11a2 Nc2 11a22 Nc 11a22 + n2G − − − − − − 1458000 27000 2000 2160 216 240 2 ˆ 1267a1 Nc Yd 261a1 a2 9a1 λ 163a1 242 2 as CF TF Nc + + + − − 405 8000 800 50 7200 2 ˆ 437a2 Nc Yd 157a2 Nc Yu 2827a1 Nc Yu 843a1 Yl 3401a2 3a2 λ − − + + − − 7200 800 960 10 480 96 2 2 2 ˆ 17Nc Yd 29 3λ 59 2 543a2 Yl 17 − as CF Nc Yd − as CF Nc Yu − + + N Yd Yu − 160 60 60 5 360 180 c 101Nc2 Yu2 157Nc Yd Yl 61Nc Ydd 199Nc Yl Yu Nc Yud 113Nc Yuu + + + + + + 360 180 240 180 24 240 2 33Yl 87Yll + + , (A.1) 40 80
– 13 –
Zα2
43 Nc 1 1 + nG − = 1 + a2 ǫ 3 3 6 2 1 2 a2 Nc 2a2 Nc a2 1849a2 43a2 Nc 43a2 + a2 2 n G + + − + nG − + ǫ 9 9 9 9 9 36 1 a1 Nc 3a1 49a2 Nc 49a2 as CF Nc + nG + + + + ǫ 120 40 24 24 2 3a1 259a2 Nc Yd Nc Yu Yl + − − − − 20 12 4 4 4 2 2 1849a2 Nc 1849a2 43 2 2 43a22 Nc 43a22 1 2 + − + nG − a2 Nc − + a2 3 n G ǫ 36 36 18 9 18 2 3 2 2 2 2 2 2 a N a N a Nc a 79507a2 1 a Nc 13a21 + n3G 2 c + 2 c + 2 + 2 − + + 2 nG 1 27 9 9 27 216 ǫ 80 400 2 2 43 1 7a1 a2 Nc 39a1 a2 22001a2 Nc 22001a2 − − − − a2 as CF Nc − a2 Nc2 Yd − 360 40 432 432 6 6 1 a2 Nc Yd a2 Nc Yl a2 Nc Yu a2 Yl 11 2 2 − a2 Nc Yu − − − − − as CA CF Nc 6 6 6 6 6 18 2 2 2 2 11a1 Nc 7a Nc 3a 1 a1 a2 Nc a1 a2 343a22 Nc2 + n2G + 1 + 1+ a1 a2 Nc2 + + + 16200 900 200 180 18 20 216 2 2 1 1 4 a2 343a2 Nc 343a2 + + a2 as CF Nc2 + a2 as CF Nc + a2s CF TF Nc + 1 + 108 216 3 3 9 200 2 43a1 a2 a1 Nc Yd 17a1 Nc Yu 3a1 Yl 77959a2 181a2 Nc Yd − + + + + + 20 48 240 16 216 48 Nc2 Yd2 1 2 1 181a2 Nc Yu 181a2 Yl 1 + + as CF Nc Yd + as CF Nc Yu − − Nc Yd Yu + 48 48 2 2 12 6 Nc2 Yu2 Nc Yd Yl Nc Ydd Nc Yl Yu Nc Yud Nc Yuu Yl2 Yll − − − − + − − − 12 6 8 6 4 8 12 8 2 2 1 91a1 13a1 a2 Nc 13a1 a2 1 287a1 Nc + nG − − + + − a1 as CF Nc ǫ 10800 1200 720 80 180 13 133 2 1 2 2 1603a22 Nc 1603a22 + + a2 as CF Nc + a CA CF Nc − as CF Nc + 81 81 12 108 s 6 2 2 2 2 2 2 2 77a1 Nc 11a1 415a2 Nc 415a2 Nc 415a22 121a1 Nc + n2G − − − − − − 97200 5400 400 1296 648 1296 2 ˆ 187a1 a2 a1 λ 533a1 Nc Yd 593a1 Nc Yu 22 163a1 + + − − − a2s CF TF Nc + 27 4800 160 10 1440 1440 2 ˆ 17a1 Yl 667111a2 a2 λ 81a2 Nc Yd 81a2 Nc Yu 81a2 Yl − − + − − − 32 5184 2 32 32 32 2Y 2 5N 7 5 5Nc2 Yu2 7 c d ˆ2 + + Nc2 Yd Yu + − as CF Nc Yd − as CF Nc Yu − λ 12 12 24 12 24 5Nc Yd Yl 19Nc Ydd 5Nc Yl Yu 3Nc Yud 19Nc Yuu + + + + + 12 48 12 8 48 2 5Y 19Yll + l + , (A.2) 24 48
– 14 –
Zαs
1 8TF nG 11CA = 1 + as − ǫ 3 3 2 64 1 121as CA 176 2 2 − as CA TF nG + as TF nG + as 2 ǫ 9 9 9 1 11a1 TF 3a2 TF 20as CA TF + nG + + + 4as CF TF ǫ 30 2 3 2 17as CA − 2TF Yd − 2TF Yu − 3 1 704 2 512 2 3 3 1331 2 3 968 2 2 2 2 + as 3 − a C + a C TF nG − a CA TF nG + a T n ǫ 27 s A 9 s A 9 s 27 s F G 1 11a21 TF 121 43a22 TF 2492 2 2 + 2 nG − a1 as CA TF − − 11a2 as CA TF − a C TF ǫ 900 45 12 27 s A 32 32 121a21 TF Nc 11a21 TF 308 2 as CA CF TF − as TF2 Yd − as TF2 Yu + n2G + − 9 3 3 4050 150 2 88 1 2 a2 TF 1120 2 224 2 2 2 2 2 + a1 as TF + a2 TF Nc + + 8a2 as TF + a CA TF + a CF TF 45 6 6 27 s 9 s 3 a1 TF Yd 17a1 TF Yu 3a2 TF Yd 3a2 TF Yu 1309a2s CA 44 + + + + + + as CA TF Yd 6 30 2 2 27 3 4 2 44 2 + as CA TF Yu + 4as CF TF Yd + 4as CF TF Yu − TF Nc Yd − TF Nc Yd Yu 3 3 3 2 2T Y Y 2T Y Y F d l F l u − TF Nc Yu2 − − TF Ydd − + 2TF Yud − TF Yuu 3 3 3 a1 a2 TF 22 11 241a22 TF 13a21 TF 1 − + a1 as CA TF − a1 as CF TF + + nG − ǫ 180 60 45 45 36 2830 2 2 410 2 4 2 2 + 2a2 as CA TF − a2 as CF TF + a C TF + a CA CF TF − as CF TF 81 s A 27 s 3 2 2 2 1331a1 TF Nc 121a1 TF 11 11a2 TF 632 2 + n2G − − − a22 TF Nc − − a CA TF2 24300 900 36 36 81 s 176 2 89a1 TF Yd 101a1 TF Yu 31a2 TF Yd 31a2 TF Yu − as CF TF2 − − − − 27 60 60 4 4 2857 2 3 − a C − 8as CA TF Yd − 8as CA TF Yu − 2as CF TF Yd − 2as CF TF Yu 162 s A 14 7 7TF Yd Yl 7TF Yl Yu 7 + 3TF Ydd + + TF Nc Yd2 + TF Nc Yd Yu + TF Nc Yu2 + 3 3 3 3 3 − 2TF Yud + 3TF Yuu
,
(A.3)
3 1 1 11Nc − nG − − ZξB = 1 + a1 ǫ 45 5 10 1 a2 Nc 9a2 11as CF Nc 137a1 Nc 81a1 + a1 − − − − nG − ǫ 1800 200 40 40 30
– 15 –
9a1 9a2 Nc Yd 17Nc Yu 3Yl − − + + + 100 20 12 60 4 2 2 63a1 7a22 Nc 39a22 121 2 533a1 Nc 1 − + + + a CA CF Nc + a1 2 n G − ǫ 54000 2000 720 80 270 s 1507a21 Nc2 217a21 Nc 81a21 a22 Nc2 a22 Nc a22 44 2 2 − nG + + + + + + a CF TF Nc 243000 4500 1000 360 36 40 135 s 3a21 a1 Nc Yd 289a1 Nc Yu 9a1 Yl 43a22 a2 Nc Yd 17a2 Nc Yu − − − − + − − 1000 144 3600 16 40 16 80 Nc2 Yd2 11 2 17 17Nc2 Yu2 9a2 Yl 1 − as CF Nc Yd − as CF Nc Yu + + Nc Y d Y u + − 16 6 30 36 90 180 5Nc Yd Yl Nc Ydd 31Nc Yl Yu 11Nc Yud 17Nc Yuu Yl2 3Yll + + − + + + + 18 24 90 60 120 4 8 2 2 1 a1 a2 Nc 9a1 a2 137a1 as CF Nc a22 Nc 1697a1 Nc 327a1 + nG + + + + − ǫ 54000 2000 3600 400 2700 135 2 2 1 1463as CA CF Nc 11 2 2 16577a21 Nc2 9a + as CF Nc + n2G − 2 + a2 as C F Nc − 10 60 1620 90 1458000 2 2 2 2 2 2 2387a1 Nc 297a1 11a2 Nc 11a2 Nc 11a2 242 2 + + + + + + a CF TF Nc 27000 2000 2160 216 240 405 s ˆ 1267a1 Nc Yd 2827a1 Nc Yu 843a1 Yl 261a1 a2 9a1 λ 163a21 − − + + + − 8000 800 50 7200 7200 800 ˆ 437a2 Nc Yd 157a2 Nc Yu 543a2 Yl 17 3401a22 3a2 λ − − + + + + as C F Nc Y d 960 10 480 96 160 60 2 2 2 2 2 ˆ 17Nc Yd 33Yl2 3λ 59 2 101Nc Yu 29 − − Nc Y d Y u − − + as CF Nc Yu + 60 5 360 180 360 40 157Nc Yd Yl 61Nc Ydd 199Nc Yl Yu Nc Yud − − − − 180 240 180 24 113Nc Yuu 87Yll − , (A.4) − 240 80 ZξW
Nc 1 1 25 − = 1 + a2 − ξW + n G − + ǫ 3 3 6 1 a2 Nc a2 25a2 8a2 ξW a2 ξW Nc a2 ξW 2 + a2 2 a2 ξ W + n G + + + − − ǫ 3 3 2 2 3 4 1 as CF Nc a1 Nc 3a1 13a2 Nc 13a2 + nG − − − − − ǫ 120 40 8 8 2 2 11a2 ξW 113a2 Nc Yd Nc Yu Yl 3a1 a2 ξW − − + + + + − 20 2 4 8 4 4 4 1 1 5 5a2 ξW 1 2 2 3 + a2 3 − a22 ξW Nc − a22 ξW − a22 ξW Nc − 2 + nG − a22 ξW ǫ 3 3 6 6 2 2 2 2 2 2 2 2 7a2 ξW a2 Nc 89a2 ξW a2 Nc a22 43a2 Nc 43a2 2 − + + nG + + + − 18 18 6 12 18 9 18 2 2 2 1525a2 1 3a1 a2 ξW 1 a Nc 13a1 + − + a1 a2 ξW Nc + + 2 nG − 1 72 ǫ 80 400 120 40
– 16 –
a2 ξ 2 a1 a2 Nc 3a1 a2 1 2 2 47 47a22 ξW + + a2 ξW Nc + 2 W + a22 ξW Nc + 120 40 6 6 24 24 1 1 11 2 4273a22 Nc 4273a22 + + a2 as CF ξW Nc + a2 as CF Nc + as CA CF Nc + 432 432 2 2 18 2 2 2 2 2 2 2 11a1 Nc 7a Nc 3a 59a2 Nc 59a2 Nc 59a22 + n2G − − 1 − 1− − − 16200 900 200 216 108 216 2 3a1 a2 ξW 3a1 a2 a1 Nc Yd 17a1 Nc Yu a 4 + − − − a2s CF TF Nc − 1 + 9 200 20 20 48 240 3 2 2 2 2 2 53a2 ξW 3a1 Yl 7a2 ξW 271a2 ξW 29629a2 1 − + + − − − a2 ξ W Nc Y d 16 6 12 24 432 4 1 a2 ξW Yl 7a2 Nc Yd 7a2 Nc Yu 7a2 Yl 1 − a2 ξW Nc Yu − − − − − as CF Nc Yd 4 4 16 16 16 2 2 2 2 2 N Y 1 N Y Nc Yd Yl Nc Ydd 1 + − as CF Nc Yu + c d + Nc2 Yd Yu + c u + 2 12 6 12 6 8 2 2 Nc Yl Yu Nc Yud Nc Yuu Yl Yll 91a21 1 287a1 Nc + − + + + + + nG 6 4 8 12 8 ǫ 10800 1200 2a1 a2 Nc 3a1 a2 ζ(3) 2a1 a2 1 1 − + + a1 as CF Nc − a1 a2 Nc ζ(3) + 15 45 5 5 180 2 2a2 ξW 7025a22 Nc 7025a22 + a22 ξW Nc + 2 + 3a22 Nc ζ(3) − + 3a22 ζ(3) − 3 3 432 432 133 2 1 2 2 8 a CA CF Nc + as CF Nc − 4a2 as CF Nc ζ(3) + a2 as CF Nc − 3 108 s 6 2 2 2 2 2 2 121a1 Nc 77a1 Nc 11a1 185a2 Nc 185a22 Nc 185a22 + n2G + + + + + 97200 5400 400 432 216 432 ˆ 533a1 Nc Yd 3a1 a2 ζ(3) 11a1 a2 a1 λ 163a21 22 − − − + + a2s CF TF Nc − 27 4800 10 32 10 1440 2 2 11a2 ξW 593a1 Nc Yu 17a1 Yl 7 1 3 2 + + − a22 ξW − a22 ξW ζ(3) − − 2a22 ξW ζ(3) 1440 32 12 2 4 ˆ 79a2 Nc Yd 79a2 Nc Yu a2 ζ(3) 143537a22 a2 λ 105a22 ξW + 2 + − + + − 8 2 1728 2 96 96 2 2 79a2 Yl 7 7 ˆ 2 − 5Nc Yd − 5 N 2 Yd Yu + + as CF Nc Yd + as CF Nc Yu + λ 96 12 12 24 12 c 5Nc2 Yu2 5Nc Yd Yl 19Nc Ydd 5Nc Yl Yu 3Nc Yud 19Nc Yuu − − − − − − 24 12 48 12 8 48 5Yl2 19Yll − , (A.5) − 24 48 +
ZξG
1 CA ξG 13CA 8TF nG = 1 + as + − − ǫ 2 6 3 2 13as CA 17 1 1 2 2 2 ξ G − as C A ξG − + as 2 as CA ǫ 4 24 8 4 + nG as CA ξG TF + 2as CA TF 3
– 17 –
1 3a2 TF 11a1 TF + nG − − − 5as CA TF − 4as CF TF ǫ 30 2 2 59as CA 11 1 2 2 2 + 2TF Yd + 2TF Yu − as CA ξG − as CA ξG + 8 16 16 3 1 1 47 403a2s CA 1 3 3 3 2 3 + as 3 − a2s CA ξG + a2s CA ξG + a2s CA ξG + ǫ 8 6 48 144 5 2 2 44 2 2 16 2 2 2 2 2 2 2 + nG − as CA ξG TF − as CA ξG TF − as CA TF + as CA TF nG 3 3 9 9 2 1 11 11 43a22 TF 11a1 TF + 2 nG − + a1 as CA ξG TF + a1 as CA TF + ǫ 900 60 60 12 3 1 19 3 2 2 2 ξG TF + a2s CA ξG TF + a2 as CA ξG TF + a2 as CA TF + a2s CA 4 4 3 6 481 2 2 62 121a21 TF Nc + as CA TF + 2a2s CA CF ξG TF + a2s CA CF TF + n2G − 27 9 4050 2 2 1 a TF 200 2 32 a1 TF Yd 11a1 TF − a22 TF Nc − 2 − as CA TF2 − a2s CF TF2 − − 150 6 6 27 9 6 17a1 TF Yu 3a2 TF Yd 3a2 TF Yu 7 13 3 3 3 2 − − − + a2s CA ξG + a2s CA ξG 30 2 2 48 24 3 7957a2s CA 143 2 3 as CA ξG − − as CA ξG TF Yd − as CA ξG TF Yu − as CA TF Yd − 96 864 2 4 − as CA TF Yu − 4as CF TF Yd − 4as CF TF Yu + TF Nc Yd2 + TF Nc Yd Yu 3 3 2T Y Y 2T Y Y 2 F F u d l l 2 + TF Ydd + − 2TF Yud + TF Yuu + TF Nc Yu + 3 3 3 1 a1 a2 TF 22 319 13a21 TF + nG + − a1 as CA TF ζ(3) + a1 as CA TF ǫ 180 60 15 360 11 241a22 TF 29 + a1 as CF TF − − 6a2 as CA TF ζ(3) + a2 as CA TF + a2 as CF TF 45 36 8 4 2 2 911 2 2 2 2 + as CA ξG TF + 12as CA TF ζ(3) − as CA TF − 16a2s CA CF TF ζ(3) 3 27 4 2 2 11 1331a21 TF Nc 121a21 TF 5 2 2 + + a22 TF Nc − as CA CF TF + as CF TF + nG 27 3 24300 900 36 2 11a2 TF 304 2 176 89a1 TF Yd 101a1 TF Yu + + as CA TF2 + + a2s CF TF2 + + 36 27 27 60 60 7 1 11 31a2 TF Yd 31a2 TF Yu 3 3 3 2 3 2 + − a2s CA ξG − a2s CA ξG ζ(3) − a2s CA ξG + 4 4 96 16 32 3 9965a2s CA 167 2 3 3 25 1 3 3 ξG ζ(3) − as CA ξG − a2s CA ζ(3) + + as CA TF Yd − a2s CA 4 96 16 864 6 25 7 14 + as CA TF Yu + 2as CF TF Yd + 2as CF TF Yu − TF Nc Yd2 − TF Nc Yd Yu 6 3 3 7T Y Y 7T Y Y 7 F u F d l l − 3TF Ydd − − TF Nc Yu2 − 3 3 3 + 2TF Yud − 3TF Yuu
.
(A.6)
– 18 –
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