Condensed Matter, Particle Physics and Cosmologyfjeger/Krakow_3.pdf · F. Jegerlehner, IFJ-PAN, Krakow Lectures 2014, — nx Lect. 3 xo 147. A summary and outlook of Lecture 2 The

Lecture 3:Parameters and Renormalization Group issues in the SM

Outline of Lecture 3:¯ The SM renormalization group equations

o SM RG running at NNLO° Matching conditions for MS parameters in terms of physical parameters

o Matching relation for the effective Fermi constanto The issue of tadpoles and non-decoupling of heavy particleso Matching and running of the top masso The top Yukawa coupling from form Mt

o The Higgs self-coupling form MH

F. Jegerlehner, IFJ-PAN, Krakow Lectures 2014, —≪x Lect. 3 x≫ 147

http://www-com.physik.hu-berlin.de/~fjeger/Krakow_2.pdf


A summary and outlook of Lecture 2

The SM as a LEET and the impact of the LEESM scenario

The SM provides a rather compact and fairly simple theory of elementaryparticles with peculiar properties like interactions mediated by gaugeinteraction related to local S U(3)c ⊗ S U(2)L ⊗ U(1)Y → S U(3)c ⊗ U(1)emnon-Abelian times Abelian structure. Speculative extension like GUTs,SUSY-GUTs, sequential Fermions, Strings, String-Compactification etc.have no phenomenological support.

Most of the peculiar properties of the SM are supported by a LEET scenario:renormalizability, Yang-Mills interaction structure, fermion-family structure,the need for the Higgs etc.




Remember: the electroweak SM has been obtained as the�

�

�

�minimal renormalizable extension

of the phenomenologically established QED and non-renormalizable weakcharged current Fermi theory. What a wealth of predictions have been madeat the beginning of the 1970’s which have turned out to be reality while noother stuff has been found in 45 years! 3rd family was a surprise, masseswere unknown

v the main impact of the LEESM hypothesis is that the bare “theory” is phys-ical, the cut-off is physical and finite and one is able to calculate the bareparameters from the phenomenologically known renormalized ones.

v as we expect/assume the cut-off to be the Planck scale: all non-renormalizable effects are dramatically suppressed at low energies includ-ing any existing and future accelerator energy (except from possible cos-mic accelerators)




v as the Planck scale also sets the scale for the cosmic Big Bang event,we actually get access to early cosmology which is governed by shortdistance physics there, i.e. by the bare system the effective parametersof which we have access to by calculating the bare parameters from theknown low energy parameters.

v as we will see in such scenario the Higgs system will play a major role inthe early universe, provided the calculations of the effective parametersactually is possible up to the Planck scale (which turns out to be a matterof the existence of an appropriate window in the SM parameter space)




l On SM parameters and the status of the SM

The attempt of an extrapolation of the SM to very high energy scalesrequires a very precise knowledge of the SM and its parameters. In thefollowing section I will give a brief overview about the SM status afterLEP/SLD, Tevatron and the first results from the LHC.




On SM parameters

❏ Motivation: old – the quest for gauge unification

Precise SM predictions require to determine the U(1)Y ⊗ SU(2)L ⊗ SU(3)c

SM gauge couplings αem, α2 and αs ≡ α3 (QCD) as accurately as possible

**** a theory can not be better than its input parameters ****

⇒ precision limitations due to non-perturbative hadronic contributions ⇐

❖ beyond SM physics gauge coupling unification?

F. Jegerlehner HU Berlin – July , 2013 –

1

On SM parameters

Do we need new physics? Stability bound of Higgs potential in SM:

SM Higgs remains perturbative up to scale Λ if it is light enough (upper bound=avoiding

Landau pole) and Higgs potential remains stable ( λ > 0 ) if Higgs mass is not too light

[parameters used: mt = 175[150− 200] GeV ; αs = 0.118]

Riesselmann, Hambye 1996

Key object of our interest:

the Higgs potential☛✡

✟✠V = m2

2H2 + λ

24H4


2

On SM parameters

❏ Motivation: new – running couplings and the Higgs vacuum sta bility

With Higgs from LHC: self-consistency of SM up to the Planck scale: Higgs vacuum stability, origin of Higgs

mechanism etc

The SM dimensionless couplings in the MS scheme as a function of the renormalization scale. The input parameter

uncertainties as given by the line thickness. The green band corresponds to Higgs masses in the range [124-127] GeV.


1

On SM parameters

The Parameters of the Standard Model

− in four fermion and vector boson processes −original focus at LEP times

our focus

g1 = g′, g2 = g ⇔ αem , sin 2Θeff

Thomsonscattering

pp, e+e−

pp, e+e−

µ–decay

e+e− → ff

e+e− → e+e−

νe, νN

α

MW

MZ

Gµ

vf , af

sin2 ΘW

SU(2)L ⊗ U(1)Y

Higgs mechanism

g2, g1, vunlike in QED and QCD in SM (SBGT)

parameter interdependence

☞

only 3 independent quantities

(besides fermion masses and mixing parameters)

α, Gµ, MZ

⇓parameter relationships between very precisely measurable quantities

precision tests, possible sign of new physics


1

On SM parameters

First basic constraints via virtual effects on top Yukawa ( mt and coupling yt) and Higgs

potential ( mH and self-coupling λ) ⇒ top at Tevatron, Higgs at LHC confirmed.✗

✖

✔

✕Now in the focus interplay between

gauge couplings g1, g2, g3, top Yukawa coupling yt and Higgs self-coupling λ

Test of quantum effects

Prototype QED:

γ γ γ

γ

µ

µ

vacuum polarization→ Lamb shiftαem shift (6%)

form factors→ anomalous magnetic

moment (99.6%)

LEP/SLC version of g − 2:

√2GµM

2Z sin2 Θi cos

2 Θi = πα (1 + δi)

sin2 Θi = (1− vℓ/aℓ)/4, 1−M2W/M2

Z , e2/g2, · · · differ by quantum corrections only

SM: renormalizable theory (’t Hooft 1971)


1

v News from LHC: Higgs found, last essential free parameter fixedMH = 125.5 ± 1.5 GeV

v all SM parameters rather precisely known now

MZ = 91.1876(21) GeV, MW = 80.385(15) GeV, Mt = 173.5(1.0) GeV,

GF = 1.16637 × 10−5 GeV−2, α−1 = 137.035999, α(5)s (M2

Z) = 0.1184(7).

Precision predictions:

sin2 Θ f , v f , a f ,MW ,ΓZ,ΓW , g2 · · ·

all depend on α effective (running αem(s))!




Electroweak fits

New physics sensible observable is the W mass given by

M2W

1 − M2W

M2Z

=πα√

2GF

(1 + ∆r) ; ∆r = f (α,GF,MZ,mt, · · · )

l ∆r model-dependent radiative corrections

l in SUSY models MW is sensitive to the top/stop sector parameters

while

sin2 Θeff =14

(1 − Re

3eff

aeff

)

remains much less affected Buchmuller et al., Heinemeyer et al.




r for MW SUSY looks favored by data!l sensitivity to top/stop sector strongly enhances in MW

l General: MSSM results merge into SM results for larger SUSY masses,as decoupling is at work.




�

�

�

�The SM fits nature!




On should keep in mind that applying the leading corrections only α(M2Z),

αs(M2Z) and the leading top correction in the ρ-parameters, the remaining

subleading corrections of various origin (self-energy, vertex, box) amount to10 σ effects, which, if not accounted for, would rule out the SM highly!

r SUSY does not yield better global fit! Means SUSY effects on precisionobservables are small, looks like heavier SUSY spectrum favored(decoupling at work)! Higgs at 125 GeV also looks to point in this direction.And the muon g − 2 deviation? A puzzle yet to be solved!




Most serious persisting 3 − 4σ deviation the Muon g − 2

constraints from LEP, B-physics, g-2, cosmic relict density [plots Olive 09].

m0 scalar mass m1/2 gaugino mass




After first LHC results:




Kazakov, de Boer et al.

l with the Higgs found at 125 GeV muon g − 2 looks to me in trouble.If δaµ is not SUSY, what else?Most other NP scenarios give likely even smaller contributions!At least GUT-SUSY scenarios likely cannot explain it!




Many open questions in the Standard Model

Standard Model very good description– Less satisfactory explanations

5 Why 3 families of fundamental fermions ?

5 Why P, C and CP violations? Why not more CP ?5 Strong CP problem ?5 Why us ? Origin of matter–antimatter asymmetry ?5 What Dark Matter ?5 Why the Cosmological Constant is so tiny ?




SM parameters are running parameters!

+ −

+−+

−

+−

+−

+− +

−

+−

+−

+−

+−

+−

+−

+−

+−

+ −r

−

−

γ virtual

pairs

γ

µ− µ−

µ− µ−

γ∗ → e+e−, µ+µ−, τ+τ−, uu, dd, · · · → γ∗

↑ tLHC ∼ 1.66 × 10−15 secIn a collision of impact energy E the effective charge is the charge contained

in the sphere of radius r ' 1/E, which due to vacuum polarization is largerthan the classical charge seen in a large sphere (r → ∞)

⇒ charge screening (charge renormalization) running charge.electromagnetic interaction strength decreases with distance

Running parameters αem and αs

r QED Abelian U(1)em




A compilation of the QCD coupling αs measurements. The lowest pointshown is at the τ lepton mass Mτ = 1.78 GeV where αs(Mτ) = 0.322 ± 0.030

↑ tLEPI ∼ 2.58 × 10−11 sec

r non-Abelian gauge theories responsible for weak [local S U(2) symmetry]and strong [local S U(3) symmetry] interactions are anti-screening≡ Asymptotic Freedom (AF) Gross, Wilczek, Politzer 1973 [NP 2004]

strong interaction strength increases with distance




The SM running parameters

The SM dimensionless couplings in the MS scheme as a function of therenormalization scale for MH = 124 − 127 GeV.




¯ The SM renormalization group equations

References:1-loop and 2-loop: Gross, Wilczek, Politzer 1973, Jones, Caswell 1974,Tarasov, Vladimirov 1977, Jones 1982, Fischler, Oliensis 1982, Machacek,Vaughn 1983/84/85, Luo, Xiao 20033–loop QCD: Tarasov, Vladimirov, Zharkov 1980, Larin, Vermaseren 19934–loop QCD: Ritbergen, Vermaseren, Larin 1997, Czakon 20052–loop QCD OS vs MS mass: Gray, Broadhurst, Grafe, Schilcher 1990,Fleischer, F.J., Tarasov, Veretin 1999/20003–loop QCD OS vs MS mass: Chetyrkin, Steinhauser 2000, Melnikov,Ritbergen 2000β(3)g′, β(3)

g :Mihaila, Salomon, Steinhauser 2012, Bednyakov, Pikelner, Velizhanin 2012β(3)yt , β

(3)λ :

Chetyrkin, Zoller 2012/2013, Bednyakov, Pikelner, Velizhanin 2012/2013




As µd/dµ = d/d ln µ it is convenient to use the dimensionless variablet = ln µ/mZ, µ the renormalization scale, mZ the Z boson mass. The RGequations for the relevant couplings may be written as

ddtg′(t) = κβ(1)

g′+ κ2β(2)

g′+ κ3β(3)

g′,

ddtg(t) = κβ(1)

g + κ2β(2)g + κ3β(3)

g ,

ddtg3(t) = κβ(1)

g3+ κ2β(2)

g3+ κ3β(3)

g3,

ddtyt(t) = κβ(1)

yt+ κ2β(2)

yt+ κ3β(3)

yt,

ddtλ(t) = κβ(1)

λ + κ2β(2)λ + κ3β(3)

λ .

The powers of the prefactor κ = 1/(16π2) represents the loop order of theβ-function coefficients.




o SM RG running at NNLOThe β-functions

1-loop

β(1)g′

=416g′3 ,

β(1)g = −

196g3 ,

β(1)g3

= −7g33 ,

β(1)yt

=92y3

t −94g2yt − 8g2

3yt −1712g′2yt ,

β(1)λ =

274g4 +

92g′2g2 − 9λg2 +

94g′4 − 36y4

t + 4λ2 − 3g′2λ + 12y2t λ .




2-loop

β(2)g′

=443g2

3g′3 +

(92g2 +

19918

g′2 −176y2

t

)g′3 ,

β(2)g = 12g2

3g3 +

(356g2 +

32g′2 −

32y2

t

)g3 ,

β(2)g3

= g33

(92g2 − 26g2

3 +116g′2 − 2y2

t

).




β(2)yt

= yt

[−108g4

3 + 9g2g23 +

199g′2g2

3 + 36y2t g

23 −

34g′2g2

−234g4 +

1187216

g′4 − 12y4t +

λ2

6+ y2

t

(22516

g2 +13116

g′2 − 2λ)],

β(2)λ = 80g2

3y2t λ − 192g2

3y4t +

9158g6 −

2898g′2g4 −

272y2

t g4

−738λg4 −

5598g′4g2 + 63g′2y2

t g2 +

394g′2λg2 − 3y4

t λ

+452y2

t λg2 −

3798g′6 + 180y6

t − 16g′2y4t −

263λ3 −

572g′4y2

t

−24y2t λ

2 + 6(3g2 + g′2

)λ2 +

62924

g′4λ +856g′2y2

t λ .




3-loop β-functions of the gauge couplings

β(3)g′

=1315

64g4g′3 − g2g2

3g′3 + 99g4

3g′(t03 +

20596

g2g′5 −13727

g23g′5

−388613

5184g′7 −

78532

g2g′3y2t −

293g2

3g′3y2

t −2827288

g′5y2t +

31516

g′3y4t ,

β(3)g =

3249531728

g7 + 39g5g23 + 81g3g4

3 +29132

g5g′2 −13g3g2

3g′2

−5597576

g3g′4 −72932

g5y2t − 7g3g2

3y2t −

59396

g3g′2y2t +

14716

g3y4t ,

β(3)g3

=109

8g4g3

3 + 21g2g53 +

652g7

3 −18g2g3

3g′2 +

779g5

3g′2

−2615216

g33g′4 −

938g2g3

3y2t − 40g5

3y2t +

10124

g33g′2y2

t + 15g33y

4t .




The SM gauge couplings running up to MPlanck




3-loop β-functions of yt and λ in gaugeless limit g′ = g = 0

β(3)yt

= 2[(−

20833

+ 320 ζ3)g63 + (

382712− 114 ζ3)g4

3y2t −

1572g2

3y4t

+(33916

+274ζ3)y6

t +43g2

3y2t λ +

332y4

t λ +596y2

t λ2 −

112λ3

],

β(3)λ = 12

[(−

2663

+ 32 ζ3)g43y

4t + (−38 + 240 ζ3)g2

3y6t − (

15998

+ 36 ζ3)y8t

+16

(1244

3− 48 ζ3)g4

3y2t λ +

16

(895 − 1296 ζ3)g23y

4t λ +

16

(1178− 198 ζ3)y6

t λ +1

36(−1224 + 1152 ζ3)g2

3y2t λ

2 +

136

(1719

2+ 756 ζ3)y4

t λ2 +

9724y2

t λ3 +

11296

(3588 + 2016 ζ3)λ4],

where ζ3 = 1.20206... is the Riemann zeta function.




Comparing 2–loop, 3–loop gaugeless and 3-loop full running of λChetyrkin, Zoller 2013

Important fact: in gaugeless limit using my MS input parameters leads to anunstable Higgs potential, while accounting for the complete calculation Ifind a stable vacuum up to MPlanck.




° Matching conditions for MS parameters in terms of physicalparameters

v we want to solve the RG equations up to very high scales, where mass ef-fects are supposed to be negligible, in regions where physical thresholdsplay no role. The mass independent MS scheme is the most simple choiceand under these circumstances likely a rather physical one (reflects thetrue UV structure of the SM, quasi-bare quantities).

v in order to solve the MS RG equations we need the input MS values of the

basic couplings g′, g, g3, yt, λ. While g3(M2Z) =

√4 παs(M2

Z) is a standard MSreference parameter, the key parameters yt and λ are known actually onlythrough the related masses Mt and MH.

v therefore the relation between the renormalized on-shell masses and theirMS version provides natural matching conditions




The relation between the MS and the OS scheme we find by writing therelation between the renormalized and the bare masses (in the bareLagrangian) in the two schemes: for bosons

m2b bare

OS= M2

b ren + δM2b

MS= m2

b ren + δm2b

=⇒ m2b ren = M2

b ren +(δM2

b − δm2b

)i.e. the mass shift is given by the MS finite part prescription applied to the(UV singular) OS mass counterterm:

m2b

∣∣∣MS − M2

b

∣∣∣OS = δM2

b

∣∣∣2ε−γ+ln 4π+ln µ2

0→ln µ2 .

for bosons andm f

∣∣∣MS − M f

∣∣∣OS = δM f

∣∣∣2ε−γ+ln 4π+ln µ2

0→ln µ2 .

for fermions.




Explicit SM one-loop expressions Fleischer, F.J. 1981

δMt = Mt C(−

169

s2W M2

W + 2M4

Z

M2H

+ 4M4

W

M2H

−1

18M2

Z −299

M2W +

169

M4W

M2Z

+A0(MH) + A0(MW) (12

+ 6M2

W

M2H

−12

M2b

M2t−

M2W

M2t

)

+A0(MZ) (3M2

Z

M2H

−1718

M2Z

M2t

+209

M2W

M2t−

169

M2W

M2Z

M2W

M2t

) + A0(Mt) 4 s2W (

43

M2W

M2t

)

+A0(Mt) (1 +1718

M2Z

M2t−

209

M2W

M2t

+169

M2W

M2Z

M2W

M2t

) + A0(Mb) (12

+12

M2b

M2t

+M2

W

M2t

)

+B0(Mt,MH,M2t ) (2 M2

t −12

M2H)

+B0(Mt,MZ ,M2t ) (−

718

M2Z −

1718

M4Z

M2t

+409

M2W +

209

M2W

M2Z

M2t−

329

M4W

M2Z

−169

M4W

M2t

)

+B0(Mb,MW ,M2t ) (

12

M4b

M2t− M2

b +12

M2t +

12

M2W

M2t

M2b +

12

M2W −

M4W

M2t

)

∑fs

A0(m f ) (−4m2

f

M2H

))




δM2H = C

(A0(MH) (3 M2

H)

+A0(MZ) (M2H + 6 M2

Z)

+B0(MH,MH,M2H) (

92

M4H)

+B0(MZ ,MZ ,M2H) (

12

M4H − 2 M2

H M2Z + 6 M4

Z)

+A0(MW) (2 M2H + 12 M2

W)

+B0(MW ,MW ,M2H) (M4

H − 4 M2H M2

W + 12 M4W)

+∑fs

[A0(m f ) (−8 m2

f ) + B0(m f ,m f ,M2H) (2 M2

H m2f − 8 m4

f )])

A0(m) = −m2(Reg + 1 − ln m2)

B0(m1,m2; s) = Reg −∫ 1

0dz ln(−sz(1 − z) + m2

1(1 − z) + m22z − iε) ,

are the basic scalar integrals with

Reg =2ε− γ + ln 4π + ln µ2

0 ≡ ln µ2 ; C =

√2 Gµ

16 π2 ; c2W =

M2W

M2Z

; s2W = 1 − c2

W




The on-loop corrections give the dominant contribution in the matchingrelations. Two-loop results are partially known F.J., Kalmykov, Veretin2002/.../2004. The expressions include the tadpole contributions andtherefore are gauge invariant and have the proper UV singularity structure.




o Matching relation for the effective Fermi constant

On-shell counterterms:Using the relation

√2Gµ = v−2 = e2

4M2

ZM2

W

1M2

Z−M2W

one may write

δGFGF

= 2δv−1

v−1

δv−1

v−1 =δee−

1

2 s2W

(s2WδM2

W

M2W

+ c2WδM2

Z

M2Z

)δee

= C s2W M2

W

(383

+ 14A0(MW)

M2W

−83

∑fs

Q2f(1 +

A0(m f )

m2f

))




δM2Z = C

(−

23

M2H M2

Z + 4M4

Z

M2H

M2Z −

29

M4Z + 8

M4W

M2H

M2Z −

209

M2W M2

Z +163

M4W − 16 c2

W M4W

+A0(MH) (13

M2H + 2 M2

Z) + A0(MZ) (−13

M2H + 6

M4Z

M2H

+23

M2Z)

+A0(MW) (12M2

W

M2H

M2Z +

163

M2W − 16 c2

W M2W +

23

M2Z)

+B0(MZ ,MH,M2Z) (−

43

M2H M2

Z +13

M4H + 4 M4

Z)

+B0(MW ,MW ,M2Z) (

163

M2W M2

Z −683

M4W − 16 c2

W M4W +

13

M4Z)

+∑fs

[(323

M2Z m2

f −169

M4Z) (a2

f + b2f )

+ A0(m f ) (−8M2

Z

M2H

m2f +

323

M2Z (a2

f + b2f ))

+ B0(m f ,m f ,M2Z) (

323

M2Z m2

f (a2f − 2b2

f ) +163

M4Z (a2

f + b2f ))

])F. Jegerlehner, IFJ-PAN, Krakow Lectures 2014, —≪x Lect. 3 x≫ 184



δM2W = C

(−

23

M2H M2

W + 8M6

W

M2H

+ 4M4

Z

M2H

M2W −

1129

M4W −

23

M2Z M2

W + A0(MH) (2 M2W +

13

M2H)

+A0(MZ) (6M2

Z

M2H

M2W +

83

M2W − 8 c2

W M2W +

13

M2Z) + A0(MW) (12

M4W

M2H

− 4 M2W −

13

M2H −

13

M2Z)

+B0(MH,MW ,M2W) (−

43

M2H M2

W +13

M4H + 4 M4

W)

+B0(MZ ,MW ,M2W) (−

683

M4W − 16 c2

W M4W +

13

M4Z +

163

M2Z M2

W)

+B0(0,MW ,M2W) (−16 s2

W M4W)

+∑fd

[43

M2W (m2

1 + m22) −

49

M4W

+ A0(m1) (−8M2

W

M2H

m21 +

43

M2W −

23

(m21 − m2

2))

+ A0(m2) (−8M2

W

M2H

m22 +

43

M2W +

23

(m21 − m2

2))

+ B0(m1,m2,M2W) (

43

M4W −

23

M2W (m2

1 + m22) −

23

(m21 − m2

2)2)])




FromGF bare = Gµ + δGF

we then obtainGMS

F = Gµ + (δGF |OS)Reg=ln µ2

Below we will argue to use µ = MZ as a matching scale.




o The issue of tadpoles and non-decoupling of heavy particlesor

tricky points in the Higgs phase of the SM

In the symmetric phase of the SM all masses except the m2–term in theHiggs potential vanish and the RG coefficient calculations and the solutionof the RG are straight forward as there are no special mass-couplingrelations and decoupling of heavy states (the four Higgs scalars) is asexpected according to the Appelquist-Carazzone theorem (AC), which inthe broken phase only holds in the QCD and QED sectors.

The Decoupling Theorem: if all external momenta of a process or in the cor-responding amplitude are small relative to the mass M of a heavy state, thenthe “light fields only” Green functions of the full theory differ from the theorywhich has no heavy fields at all, only by finite renormalizations of couplings,masses and fields of the light theory, up to terms which are suppressed byinverse powers of the heavy mass. Thus further corrections are of the form(µ/M)x with x ≥ 1.




It means that only the renormalization subtraction constants are de-pendent on M (logarithms) and this M–dependence gets renormalizedaway by physical subtraction conditions.

As a S -matrix does not exist in the symmetric phase (IR problems) there isno on-shell renormalization scheme and the intrinsically massless MSscheme is the natural choice.

All couplings, the gauge couplings g′, g, gs, the Yukawa couplings y f , as wellas the Higgs self-coupling λ are dimensionless and run because oflogarithmic UV behavior.

The renormalization of the Higgs potential mass term m2 (the only relevantoperator in the Lagrangian) requires to remove the quadratic divergences,and hence potentially is subject to a hierarchy problem. The scaledependence of the renormalized m2 is ruled by its mass anomalousdimension.




For m2 < 0 the Higgs mechanism takes place and implies a non-trivialvacuum characterized by a non-vanishing VEV (order parameter)

Hs = v + H ; 〈Hs〉 = v ; 〈H〉 = 0 ,

Hs is the original Higgs field of the symmetric phase H is the physical Higgsfield of mass MH. The condition

〈H〉 = 0

is not stable under renormalization, as we are in the broken phase, while anon-zero VEV is forbidden in the symmetric phase by the

discrete Z2 symmetry H → −H .




In perturbation theory contributions to 〈H〉 are the tadpoles :

〈H〉 = + = 0

ΠµνV (k2) ≡ +

Σf (k2) ≡ +

ΠH(k2) ≡ +

and appear as a contribution to the irreducible self-energies of all massiveparticles.




At one loop in the SM the tadpole counterterm needed to adjust 〈H〉 = 0 is

δvtp = v1

16π2v2

1M2

H

[−2M4

Z − 4M2W + A0(MH) (−

32

M2H) + A0(MZ) (−3 M2

Z)

+A0(MW) (−6 M2W) + A0(

√ξMZ) (−

12

M2H) + A0(

√ξMW) (−M2

H)

+∑

f

4 M2f A0(M f )

]where ξ is the gauge parameter in the linear ’t Hooft gauge.

r tadpoles are UV singular

r tadpoles are not gauge invariant

Consequently, mass counter terms, like

Γ(2)H = i

[p2 − M2

H ren − δM2H + ΠH(p2)

]; δM2

H = ΠH

(M2

H ren

)F. Jegerlehner, IFJ-PAN, Krakow Lectures 2014, —≪x Lect. 3 x≫ 191



are gauge invariant only if tadpoles are included. This is a controversialpoint, because there is a theorem saying that tadpoles do never contributeto physical observables! J.C. Taylor 1976, E. Kraus 1998

The no-Tadpoles Theorem: Physical amplitudes as functions of phys-ical parameters are devoid of tadpoles.

That’s why very often in higher order calculations tadpole contributions areomitted as contributions.

Our point of view: taking into account all diagrams according to SMFeynman rules cannot be wrong. If at the end it turns out that tadpolescancel in a given calculation the result remains the same.

At the end the upshot of SBGT is thatthe UV renormalization structure is not affected by spontaneous SB

and thus identical in both the symmetric and the broken phase.




Omitting tadpoles implies:à the relations between bare and renormalized parameters get messed up,

UV structure lost, gauge invariance lost

à as MS quantities are directly extracted from the bare ones, also therelationship between MS and on-shell quantities are affected.Such pseudo MS quantities do not satisfy the correct RG equationand they are gauge dependent

à the RG equations in the symmetric phase, where all counterterms aredevoid of tadpoles, can be reproduced from the mass counterterms,calculated in the broken phase, via the bare mass-coupling relations:

m2W(µ2) = 1

4 g2(µ2) v2(µ2) ; m2

Z(µ2) = 14 (g2(µ2) + g′2(µ2)) v2(µ2) ;

m2f (µ

2) = 12 y

2f (µ

2) v2(µ2) ; m2H(µ2) = 1

3 λ(µ2) v2(µ2) .

which tell us that at fixed v a particle can be heavy only in the correspondingstrong coupling limit, in which large effects are unavoidable.




From these relations we derive the key relations

µ2 ddµ2v

2(µ2) = 4 µ2 ddµ2

m2W(µ2)g2(µ2)

= 4 µ2 ddµ2

m2Z(µ2) − m2

W(µ2)g′2(µ2)

= 3 µ2 d

dµ2

m2H(µ2)λ(µ2)

= 2 µ2 ddµ2

m2f (µ

2)

y2f (µ

2)

= v2(µ2)[γm2 −

βλλ

],

which relates the RG for the order parameter v to the parameters m and λ ofthe Higgs potential. Surprising or not the β-function related to v exhibitsnon-analytic terms in λ of the form βλ

λ. The terms originate from the 1

M2H

(zero

momentum transfer Higgs propagator) factor of the contributing tadpoles4.

Ü lesson 1: tadpoles can produce large shifts in the relations betweenthe “quasi-bare” MS parameters and the on-shell ones and are neces-sary to preserve gauge invariance and the correct RG equations

4In the pure Higgs system, where β(λ) ∝ λ there would be no problem because βλ/λ would be regular at λ = 0. Inthe SM βλ|λ=0 = βλ(g, g′, gs, yt, λ = 0) , 0 and we would have a true pole.




For the weak sector of the SM is important that, besides the tadpole issue,that the AC-theorem does not apply:

l the non-decoupling in the weak sector of the SM is a consequence of themass coupling relations, which hold if the masses are generated by theHiggs mechanism

l in QCD the MS scheme is utilized in conjunction with “decoupling byhand” considering an effective ’n f active flavors’ QCD to be matched atsuccessive flavor thresholds

l there is no decoupling for the W and Z bosons: the limit MW → ∞ can beachieved by letting g→ ∞ or v→ ∞ or both. In nature, only the limit g→ ∞leads to the observed low-energy limit of the effective four-fermion theorywith

√2 Gµ = 1/v2 fixed, by nuclear β–decay etc. This obviously is a

non-decoupling effect. In contrast to QED or QCD, the low-energy effectivetheory (obtained after elimination of the heavy state) is a non-renormalizableone exhibiting a completely wrong high-energy behavior




l most well-known non-decoupling effects due to heavy top mass

à leading correction in EW ρ parameter

ρ = GNC/GCC(0) = 1 +NcGµ

8π2√

2

m2t + m2

b −2m2

t m2b

m2t − m2

b

lnm2

t

m2b

≈ 1 +Ncy

2t

32π2 ,

measures weak-isospin breaking by the Yukawa couplings of the heavyfermions at zero momentum.

1995 LEP bound on Mt Ü 1995 Top discovery at Tevatron

Ü lesson 2: “decoupling by hand” cannot be applied for weak con-tributions, i.e., we cannot parametrize and match together effectivetheories by switching off fields of mass M > µ at a given scale µ.

l potentially, the Higgs VEV v, which determines the Fermi constant viaGF = 1

√2v2

, could be particularly affected by non-decoupling effects




However, we may take advantage of the fact that tadpole contributions dropout from relations between physical (on-shell) parameters and amplitudes:

o compare:

à low energy – Gµ determined by the muon lifetime

à W mass scale – Gµ =12πΓW`ν√

2M3W

given by leptonic W decay rate.

Indeed: Gµ ≈ Gµ with good accuracy is not surprising because the tadpolecorrections which potentially lead to substantial corrections are absent inrelations between observable quantities as we know.

o consequently a SM parametrization in terms of α(MZ), αs(MZ),Gµ and MZ

(besides the other masses), provides a good parametrization of theobservables extracted from experiments at the vector boson mass scale(LEP precision physics).

o While α(MZ) and αs(MZ) are strongly scale dependent, Gµ remains




unrenormalized below the Z mass scale, which thus is an ideal matchingscale, to evaluate the MS parameters in terms of corresponding on-shellvalues.

r Usually the scale independence of the effective GF below the W massscale is “explained” by a “decoupling by hand” argument by inspection ofthe one-loop RG equation

µ2 d ln GMSF

dµ2 =

√2 GMS

F

8π2

{∑f

(m2f − 4

m4f

m2H

) − 3 m2W + 6

m4W

m2H

− 3/2 m2Z + 3

m4Z

m2H

+ 3/2 m2H

},

If we sum terms only over m < µ, there is effectively no running, because ofthe smallness of the light fermion masses. GF is expected to start runningonce MW, MZ, MH and Mt come into play. At higher scales, certainly, the MSversion of v(µ2) or equivalently GMS

F (µ2) must be running as required by thecorresponding RG.




o Matching and running of the top mass

r quark masses in the MS scheme

mq,bare = mq(µ2)[1 + αs

∑i=0

αis

i+1∑k=1

δZ(i,k)αs

.k+ α

∑i, j=0

αiαjs

i+ j+1∑k=1

δZ(i, j,k)α

.k

].

incl. QCD corrections, EW contribution mixed in higher orders with QCD

r quark mass anomalous dimension

µ2 ddµ2 ln m2

q = γq(αs, α) =

[αs

∂

∂αs+ α

∂

∂α

] [αs

∑i=0

αisδZ

(i,1)αs

+α∑i, j=0

αiαjsδZ

(i, j,1)α

],

split into the pure QCD plus all others, which we call EW contributionsγq(αs, α) = γQCD

q + γEWq , the EW contribution to the fermion anomalous

dimension γEWt in the MS scheme may be written in terms of




RG functions of parameters in the unbroken phase of the SM asγEW

t = γyt + 12γm2 −

12βλλ,

where γm2 = µ2 ddµ2 ln m2, βλ = µ2 d

dµ2λ, γyq = µ2 ddµ2 ln yq, yq is the Yukawa coupling

of quark q, and m2 and λ are the parameters of the scalar potentialV = m2

2 φ2 + λ

24φ4.

r RG equation for Higgs VEV v(µ2) follows

µ2 ddµ2v

2(µ2) = v2(µ2)[γm2 −

βλλ

],

r the anomalous dimension of GMSF is γGF = −

[γm2 −

βλλ

],

Note: the anomalous dimension of v2(µ2) adopted here via diagrammaticcalculations differs from the anomalous dimension of the scalar field asobtained in the effective-potential approach.




r matching conditions between pole and running masses

Mt − mt(µ2) = mt(µ2)∑j=1

(αs(µ2)π

) j

ρ j + mt(µ2)∑

i=1; j=0

(α(µ2)π

)i (αs(µ2)π

) j

ri j .

The QCD corrections ρ j ( j = 2, 3) Gray, Broadhurst, Grafe, Schilcher 1990;Fleischer, F.J., Tarasov, Veretin 1999; Chetyrkin, Steinhauser 1999/2000;Melnikov, Ritbergen 2000 r10 Hempfling, Kniehl 1995 and r11 F.J., Kalmykov2003/2004 r12 in the gaugeless-limit Martin 2005




Numerical result for mt − Mt

F.J., Kalmykov, Kniehl 2012

{mt(M2

t ) − Mt

}QCD

= Mt

[−

43α(6)

s (M2t )

π− 9.125

α(6)s (M2

t )π

2

− 80.405

α(6)s (M2

t )π

3].

Using α(6)s (M2

t ) = 0.1079(6), which follows from the value of α(5)s (M2

Z) viafour-loop evolution and three-loop matching we obtain the numerical result{

mt(M2t ) − Mt

}QCD

= −7.95 GeV − 1.87 GeV − 0.57 GeV = −10.38 GeV .




Numerical results for the difference mt(M2t ) − Mt

The two-loop corrections of order O(α2) are not yet. The largest contributionis coming from tadpole diagrams, which are known and allows us toestimate the error due to the unknown higher-order corrections, which isabout 1 GeV.




The various contributions to mt(M2t ) − Mt in GeV.

MH [GeV] O(α) O(ααs) O(α) + O(ααs) O(αs) + O(α2s) + O(α3

s) total124 12.11 −0.39 11.72 −10.38 1.34125 11.91 −0.39 11.52 −10.38 1.14126 11.71 −0.38 11.32 −10.38 0.94

As a result, we observe a large EW correction, which for the assumed MH

range by accident almost perfectly cancels the QCD correction. The relationmt(M2

t ) vs. Mt can be parametrized as

{mt(M2

t )−Mt

}SM

={mt(M2

t )−Mt

}QCD

+

[0.0664−0.00115 ×

( MH

1 GeV−125

)]Mt .

Our calculation shows that the large leading correction, of O(α), to the shiftmt(M2

t ) − Mt is not substantially modified by the next-to-leading term, ofO(ααs). Radiative corrections beyond the presently known ones are likely tobe small and not to change the observed quenching qualitatively.




o The top Yukawa coupling from form Mt

Once again:

In the weak sector of the SM, there is no decoupling because massesand couplings are interrelated by the Higgs mechanism. So “decou-pling by hand” as usually applied in QCD by considering an effective‘n f active flavors’ QCD to be matched at successive flavor thresholds,and which can be applied to QED as well, does not make sense in theweak sector.

On phenomenological grounds, as GF has been measured to agree at the MZ

scale with its low energy version Gµ and because Yukawa couplings run asthey do in the symmetric phase, below of the EW scale, one may defineeffective light fermion masses to run via their Yukawa couplings only:

m f (µ2) = 2−3/4G−1/2µ y f (µ2) .

l As the Yukawa couplings y f (µ2) are not affected by the Higgs mechanism,




the EW corrections to the Yukawa couplings are free of tadpoles and/orquadratic divergences.

l Since real physical observables are also free of tadpole contributions, thisproperty is an additional argument why the advocated relation is a goodcandidate for the evaluation of the EW contributions to the ratio betweenpole and MS masses of lighter quarks, such as the bottom and charm quarks

r In short: fermion masses and Yukawa couplings have equivalent RGevolution as long as GF or, equivalently, v can be taken not to be running and

identify GMSF (µ2) = Gµ.

l Alternatively, and more consequently concerning the decoupling issue,the proper MS definition of a running fermion mass is

m f (µ2) = 2−3/4(GMS

F

)−1/2(µ2) y f (µ2) , (F)




where GMSF (µ2) and m f (µ2) are solutions of

γGF = µ2 ddµ2 ln GMS

F (µ2) = −µ2 ddµ2 ln v2(µ2) = −

[γm2 −

βλλ

],

and

µ2 ddµ2 ln m f (µ2) = γQCD

q + γy f −12γGF ,

respectively. For the MS top quark mass we advocate to use (F), whichamong others includes the tadpole contributions.

à Note that the difference between the two definitions is particularlysignificant for the top quark. As both versions are gauge invariant bydefinition, the difference is not just dropping or not the tadpole terms.




The running of GMSF definitely starts at about µ ∼ 2MW

5, when the scale of aprocess exceeds the masses of the weak gauge bosons. Since the topquark is the heaviest particle in the SM, at least here the “decoupling byhand” prescription becomes obsolete and we have to take full SM parameterrelations as they are.

The top Yukawa coupling running up to MPlanck

5As the on-shell GF at the Z boson mass scale can be identified with Gµ (numerically) it is justified to match GMSF

with Gµ at the scale MZ.




o The Higgs self-coupling form MH

Note that the only information we have on the Higgs self-coupling λ is viathe experimental value of MH:

λ = 3√

2Gµ M2H.

From the physical Higgs mass MH and the relation

m2H = M2

H +(δM2

H

∣∣∣OS

)Reg=ln µ2

one may evaluate the relation between λ and λMS under the proviso thatGMS

F = Gµ ??? . In principle GMSF is expected to be a running parameter as

well. As argued above indeed up to the W boson mass scale GMSF can be

taken not to run. The result of the evaluation may be parametrized as

λ(MH) = 0.8065 + 0.0109 (MH[GeV] − 126) + 0.0015 (Mt[GeV] − 172) +0.0002−0.006 ,

central value refers to µ = MH. Fleischer, F.J. 1981, Sirlin, Zucchini 1986




2-loop matching

The QCD contribution, λ(2)QCD(µ), is obtained evaluating the relevant diagrams

via a Taylor series in xht ≡ m2h/m

2t up to fourth order

λ(2)QCD(µ) = 6 ×

G2µm

4t

(4π)4 Nc CF g23(µ)

[16

(−4 − 6LT + 3L2

T

)+ xht

(35 −

2 π2

3+ 12LT − 12L2

T

)+ x2

ht61135

+ x3ht

12236300

+ x4ht

431231323000

],

where Nc and CF are color factors (Nc = 3, CF = 4/3) and LT = ln(m2t /µ

2).Bezrukov et al 2012, Degrassi et al 2012




The Higgs self-coupling in comparison with the other couplings

With my input values for the MSb parameters (which slightly differ in thetop-Yukawa coupling) I get a stable Higgs vacuum i.e. λ(µ) remains positiveup to MPlanck such that the SM can be valid by itself up to the Planck scale.




RG behavior of the Higgs VEV

v For large values of µ2, the behavior of the running Fermi constant GF(µ2) isdefined by the Higgs self-coupling and the sign of its beta-function βλ:

µ2 ddµ2 ln GF(µ2) ∼

βλ(µ2)λ(µ2)

.

v The detailed perturbative analysis of the r.h.s. of this equation was per-formed recently and reveals that the beta function βλ is negative up to ascale of about 1017 GeV, where it changes sign. Above the zero of βλ, theeffective coupling starts to increase again, and the key question is whetherat the zero of the beta function the effective coupling is still positive. In thelatter case, it will remain positive although small up to the Planck scale.




v In any case, at moderately high scales where βλ < 0, and provided that λ isstill positive, the following behavior is valid for the Fermi constant:

GF(µ2)∣∣∣µ2→∞

∼(µ2

)βλ(µ2)λ(µ2) → 0 ,

being decreasing, which means that v2(µ2) is increasing at these scales(where βλ < 0 and λ > 0).

v Most of the recent analyses of find that λ turns negative (unstable or meta-stable Higgs potential) before the beta function reaches its zero. This mayhappen at rather low scales, at around 1010 GeV. In this case, we would getan infinite Higgs vacuum expectation value far below the Planck scale asan essential singularity.

v Given the present uncertainty in the value of Mt, there is a good chance




that λ remains positive up to the zero of the beta function and as a conse-quence up to the Planck scale.

v Then GF(µ2) would start to increase again, and v(µ2) would start to decreasebut remain finite (about 685 GeV) at the Plank scale, implying that all effec-tive masses stay bounded.

v The effectively massless symmetric phase of the SM would then be ob-tained at high energies by the fact that mass effects are suppressed fordimensional reasons: according to the RG, for a vertex function under adilatation of all momenta, {pi} → {κpi}, up to the overall dynamical dimen-sion and wave-function renormalizations, the result is given by replacinggi → gi(κ) and mi → mi(κ)/κ at fixed {pi} and renormalization scale µ. I.e.provided that m(κ)/κ → 0 as κ → ∞, the high-energy asymptotic effectivetheory is effectively massless as expected in the symmetric phase.




102 104 106 108 1010 1012 1014 1016 1018 1020

0.0

0.2

0.4

0.6

0.8

1.0

RGE scale Μ in GeV

SM

coupli

ngs

g1

g2

g3yt

Λyb

m in TeV

Renormalization of the SM gauge couplings g1 =√

5/3gY , g2, g3, of the top,bottom and τ couplings (yt, yb, yτ), of the Higgs quartic coupling λ and of theHiggs mass parameter m. All parameters are defined in the MS scheme. Weinclude two-loop thresholds at the weak scale and three-loop RG equations.

The thickness indicates the ±1σ uncertainties in Mt,Mh, α3.




At the Planck mass, Degrassi et al. 2013 find the following SM parameters:

g1(MPlanck) = 0.6168

g2(MPlanck) = 0.5057

g3(MPlanck) = 0.4873 + 0.0002α3(MZ) − 0.1184

0.0007

yt(MPlanck) = 0.3823 + 0.0051( Mt

GeV− 173.35

)− 0.0021

α3(MZ) − 0.11840.0007

λ(MPlanck) = −0.0128 − 0.0065( Mt

GeV− 173.35

)+

+0.0018α3(MZ) − 0.1184

0.0007+ 0.0029

( Mh

GeV− 125.66

)m(MPlanck) = 140.2 GeV + 1.6 GeV

( Mh

GeV− 125.66

)+

−0.25 GeV( Mt

GeV− 173.35

)+ 0.05 GeV

α3(MZ) − 0.11840.0007




Addendum: Complete NNLO RG for Higgs potential parameters λ and m2

β(1)λ

= − y4τ − y

4b3 + g4

2916

+ g21g

22

38

+ g41

316

+ λy2τ2 + λy2

b6 − λg22

92− λg2

132

+ λ212 + y2t λ6 − y4

t 3 ,

β(2)λ

=y6τ5 + y6

b15 − g42y

2τ

38− g4

2y2b

98

+ g62

(49732− 2Ng

)− g2

1y4τ2 + g2

1y4b

23

+ g21g

22y

2τ

114

+ g21g

22y

2b

94− g2

1g42

(9796

+23

Ng

)− g4

1y2τ

258

+ g41y

2b

58

− g41g

22

(23996

+109

Ng

)− g6

1

(5996

+109

Ng

)− λy4

τ12− λy4

b32

+ λg22y

2τ

154

+ λg22y

2b

454

+ λg42

(−

31316

+ 5Ng

)+ λg2

1y2τ

254

+ λg21y

2b

2512

+ λg21g

22

398

+ λg41

(22948

+259

Ng

)− λ2y2

τ24 − λ2y2b72 + λ2g2

254

+ λ2g2118 − λ3156 − y2

t y4b3 − y2

t g42

98

+ y2t g

21g

22

214− y2

t g41

198

− y2t λy

2b21 + y2

t λg22

454

+ y2t λg

21

8512− y2

t λ272 − y4

t y2b3 − y4

t g21

43

− y4t λ

32

+ y6t 15 − g2

sy4b16 + g2

sλy2b40 + g2

sy2t λ40 − g2

sy4t 16.




β(3)λ

=y8τ

(−

1438− 12ζ3

)− y2

by6τ

2978− y4

by4τ72 − y6

by2τ

2978

+ y8b

(−

15998− 36ζ3

)+ g2

2y6τ

(113732− 9ζ3

)+ g2

2y6b

(341132− 27ζ3

)+ g4

2y4τ

(4503128

−27316

ζ3 −134

Ng

)+ g4

2y2by

2τ

98

+ g42y

4b

(13653128

−81916

ζ3 −394

Ng

)+ g6

2y2τ

(−

5739256

+994ζ3 +

92

Ng

)+ g6

2y2b

(−

17217256

+2974ζ3 +

272

Ng

)+ g8

2

(9822913072

−2781128

ζ3 −14749192

Ng

−45Ngζ3 −53

N2g

)+ g2

1y6τ

(13532

+ 33ζ3

)+ g2

1y6b

(511196− 25ζ3

)+ g2

1g22y

4τ

(−

1564−

3818ζ3

)− g2

1g22y

2by

2τ

54

+ g21g

22y

4b

(−

3239192

−3118ζ3

)+ g2

1g42y

2τ

(1833256

−32ζ3 −

12

Ng

)+ g2

1g42y

2b

(4179256

+ 9ζ3 +52

Ng

)+ g2

1g62

(−

540533456

−40532

ζ3 −8341864

Ng −1027

N2g

)+ g4

1y4τ

(5697128

+37516

ζ3 +6512

Ng

)+ g4

1y2by

2τ

4124

+ g41y

4b

(151373456

−2035144

ζ3 −41536

Ng

)+ g4

1g22y

2τ

(6657256

−152ζ3 −

56

Ng

)(1)




+ g41g

22y

2b

(4403256

+92ζ3 +

256

Ng

)+ g4

1g42

(−

646933456

+87364

ζ3 +149648

Ng + 7Ngζ3

−5081

N2g

)+ g6

1y2τ

(3929256

−154ζ3 +

556

Ng

)+ g6

1y2b

(120432304

+54ζ3 +

9518

Ng

)+ g6

1g22

(−

297796912

+7532ζ3 −

180012592

Ng +619

Ngζ3 −250243

N2g

)+ g8

1

(−

68459216

+99

128ζ3 −

207351728

Ng +959

Ngζ3 −12581

N2g

)+ λy6

τ

(−

12418− 66ζ3

)+ λy2

by4τ240 + λy4

by2τ240 + λy6

b

(1178− 198ζ3

)+ λg2

2y4τ

(−

15878

+ 171ζ3

)− λg2

2y2by

2τ27 + λg2

2y4b

(−

49778

+ 513ζ3

)+ λg4

2y2τ

(−

131164−

1172ζ3 −

214

Ng

)+ λg4

2y2b

(−

393364−

3512ζ3 −

634

Ng

)+ λg6

2

(−

46489288

+2259

8ζ3 +

351536

Ng

+90Ngζ3 +709

N2g

)+ λg2

1y4τ

(5078− 117ζ3

)− λg2

1y2by

2τ9 + λg2

1y4b

(−

573724

+249ζ3)

+ λg21g

22y

2τ

(−

377132

+ 126ζ3

)+ λg2

1g22y

2b

(−

300932

+ 12ζ3

)(2)




+ λg21g

42

(455332−

2498ζ3 +

332

Ng − 6Ngζ3

)+ λg4

1y2τ

(−

178364−

1232ζ3 −

654

Ng

)+ λg4

1y2b

(−

1273031728

−476ζ3 −

15536

Ng

)+ λg4

1g22

(9798−

38ζ3 +

954

Ng − 6Ngζ3

)+ λg6

1

(12679

432+

98ζ3 +

5995162

Ng −190

9Ngζ3 +

1750243

N2g

)+ λ2y4

τ

(7172

+ 252ζ3

)− λ2y2

by2τ216 + λ2y4

b

(1719

2+ 756ζ3

)+ λ2g2

2y2τ

(2134− 144ζ3

)+ λ2g2

2y2b

(6394

−432ζ3)

+ λ2g42

(1995

8− 513ζ3 − 141Ng

)+ λ2g2

1y2τ

(−

5414

+ 96ζ3

)+ λ2g2

1y2b

(417

4− 192ζ3

)+ λ2g2

1g22(−333 − 162ζ3

)+ λ2g4

1(−183 − 81ζ3

−2353

Ng

)+ λ3y2

τ291 + λ3y2b873 + λ3g2

2(−474 + 72ζ3

)+ λ3g2

1(−158 + 24ζ3

)+ λ4 (

3588 + 2016ζ3)

+ y2t y

6τ

(−

2978

)+ y2

t y2by

4τ12 + y2

t y4by

2τ

458

+ y2t y

6b

(−

7178

−36ζ3)

+ y2t g

22y

4b

47732

+ y2t g

42y

2τ

98

+ y2t g

42y

2b

(−

35164

+117

2ζ3 − 12Ng

)(3)




+ y2t g

62

(−

17217256

+2974ζ3 +

272

Ng

)+ y2

t g21y

4b

(−

229996

+ 26ζ3

)+ y2

t g21g

22y

2τ

294

+ y2t g

21g

22y

2b

(1001

96+

312ζ3

)+ y2

t g21g

42

(3103256

+274ζ3 +

12

Ng

)+ y2

t g41y

2τ

70124

+ y2t g

41y

2b

(−

70964− ζ3

)+ y2

t g41g

22

(23521768

− 3ζ3 +56

Ng

)+ y2

t g61

(429432304

−52ζ3 +

21518

Ng

)+ y2

t λy4τ240 + y2

t λy2by

2τ21

+ y2t λy

4b

(6399

8+ 144ζ3

)− y2

t λg22y

2τ27 + y2

t λg22y

2b

(−

5314

+ 54ζ3

)+ y2

t λg42

(−

393364−

3512ζ3 −

634

Ng

)− y2

t λg21y

2τ9 + y2

t λg21y

2b

(−

92912− 2ζ3

)+ y2

t λg21g

22

(−

650932

+ 177ζ3

)+ y2

t λg41

(−

1124471728

−449

6ζ3 −

63536

Ng

)− y2

t λ2y2τ216 + y2

t λ2y2

b(117 − 864ζ3

)+ y2

t λ2g2

2

(6394− 432ζ3

)+ y2

t λ2g2

1

(−

1954− 48ζ3

)+ y2

t λ3873 − y4

t y4τ72 + y4

t y2by

2τ

458

+ y4t y

4b72ζ3

(4)




+ y4t g

22y

2b

47732

+ y4t g

42

(13653128

−81916

ζ3 −394

Ng

)+ y4

t g21y

2b

(1337

96− 28ζ3

)+ y4

t g21g

22

(−

1079192

−743

8ζ3

)+ y4

t g41

(1009133456

+2957144

ζ3 −11536

Ng

)+ y4

t λy2τ240 + y4

t λy2b

(6399

8+ 144ζ3

)+ y4

t λg22

(−

49778

+ 513ζ3

)+ y4

t λg21

(−

248524

+ 57ζ3

)+ y4

t λ2(1719

2+ 756ζ3

)− y6

t y2τ

2978

+ y6t y

2b

(−

7178− 36ζ3

)+ y6

t g22

(341132− 27ζ3

)+ y6

t g21

(346796

+ 17ζ3

)+ y6

t λ

(1178− 198ζ3

)+ y8

t

(−

15998− 36ζ3

)+ g2

sy6b(−38 + 240ζ3

)+ g2

sg22y

4b

(−

312

+ 24ζ3

)+ g2

sg42y

2b

(6518− 54ζ3

)+ g2

sg62

(−

1538

Ng + 18Ngζ3

)+ g2

sg21y

4b

(−

64118

+1363ζ3

)+ g2

sg21g

22y

2b

(233

4− 36ζ3

)+ g2

sg21g

42

(−

518

Ng + 6Ngζ3

)(5)




+ g2sg

41y

2b

(68324− 18ζ3

)+ g2

sg41g

22

(−

18724

Ng +223

Ngζ3

)+ g2

sg61

(−

18724

Ng

+223

Ngζ3

)+ g2

sλy4b(895 − 1296ζ3

)+ g2

sλg22y

2b

(−

4892

+ 216ζ3

)+ g2

sλg42

(1352

Ng − 72Ngζ3

)+ g2

sλg21y

2b

(−

99118

+ 40ζ3

)+ g2

sλg41

(552

Ng

−883

Ngζ3

)+ g2

sλ2y2

b(−1224 + 1152ζ3

)+ g2

sy2t y

4b(−2 − 48ζ3

)+ g2

sy2t g

22y

2b(−8 + 96ζ3

)+ g2

sy2t g

42

(6518− 54ζ3

)+ g2

sy2t g

21g

22

(2494− 36ζ3

)+ g2

sy2t g

41

(58724− 18ζ3

)+ g2

sy2t λy

2b(82 − 96ζ3

)+ g2

sy2t λg

22

(−

4892

+ 216ζ3

)+ g2

sy2t λg

21

(−

241918

+ 136ζ3

)+ g2

sy2t λ

2 (−1224 + 1152ζ3

)+ g2

sy4t y

2b(−2 − 48ζ3

)+ g2

sy4t g

22

(−

312

+ 24ζ3

)+ g2

sy4t g

21

(93118−

563ζ3

)+ g2

sy4t λ

(895 − 1296ζ3

)+ g2

sy6t(−38 + 240ζ3

)+ g4

sy4b

(−

6263

+ 32ζ3 + 40Ng

)(6)




+ g4sλy

2b

(1820

3− 48ζ3 − 64Ng

)+ g4

sy2t y

2b192 + g4

sy2t λ

(1820

3− 48ζ3 − 64Ng

)+ g4

sy4t

(−

6263

+ 32ζ3 + 40Ng

). (7)

β(1)m2

m2 =y2τ + y2

b3 − g22

94− g2

134

+ λ6 + y2t 3,

β(2)m2

m2 = − y4τ

94− y4

b274

+ g22y

2τ

158

+ g22y

2b

458

+ g42

(−

38532

+52

Ng

)+ g2

1y2τ

258

+ g21y

2b

2524

+ g21g

22

1516

+ g41

(+

15796

+2518

Ng

)− λy2

τ12 − λy2b36 + λg2

236 + λg2112

− λ230 − y2t y

2b

212

+ y2t g

22

458

+ y2t g

21

8524− y2

t λ36

− y4t

274

+ g2sy

2b20 + g2

sy2t 20,




β(3)m2

m2 =y6τ

(−

23316

+ 15ζ3

)+ y2

by4τ72 + y4

by2τ72 + y6

b

(160516

+ 45ζ3

)+ g2

2y4τ

(−

98716

+ 54ζ3

)− g2

2y2by

2τ

272

+ g22y

4b

(−

317716

+ 162ζ3

)+ g4

2y2τ

(−

255128−

814ζ3 −

218

Ng

)+ g4

2y2b

(−

765128−

2434ζ3 −

638

Ng

)+ g6

2

(−

39415576

+71116

ζ3 +286772

Ng + 45Ngζ3 +359

N2g

)+ g2

1y4τ

(29116− 36ζ3

)− g2

1y2by

2τ

92

+ g21y

4b

(−

106716

+ 72ζ3

)+ g2

1g22y

2τ

(−

233164

+ 45ζ3

)− g2

1g22y

2b

86564

+ g21g

42

(269164−

40516

ζ3 +214

Ng − 3Ngζ3

)+ g4

1y2τ

(−

3607128

−154ζ3 −

658

Ng

)+ g4

1y2b

(−

792073456

−3512ζ3 −

15572

Ng

)+ g4

1g22

(105332−

20716

ζ3 +558

Ng − 3Ngζ3

)+ g6

1

(839108−

5116ζ3 +

4375324

Ng −959

Ngζ3 +875243

N2g

)+ λy4

τ

(2614

+ 72ζ3

)




− λy2by

2τ108 + λy4

b

(351

4+ 216ζ3

)+ λg2

2y2τ

(1898− 108ζ3

)+ λg2

2y2b

(5678− 324ζ3

)+ λg4

2

(11511

32− 162ζ3 −

1532

Ng

)+ λg2

1y2τ

(−

5498

+ 36ζ3

)+ λg2

1y2b

(3938− 132ζ3

)+ λg2

1g22

(−

170116

+ 36ζ3

)+ λg4

1

(−

107732− 18ζ3 −

852

Ng

)+ λ2y2

τ992

+ λ2y2b

2972

+ λ2g22(−63 − 108ζ3

)+ λ2g2

1(−21 − 36ζ3

)+ λ31026 + y2

t y4τ72 + y2

t y2by

2τ

212

+ y2t y

4b

(404716

+ 36ζ3

)+ y2

t g22y

2τ

(−

272

)+ y2

t g22y

2b

(−

2438− 27ζ3

)+ y2

t g42

(−

765128−

2434ζ3 −

638

Ng

)− y2

t g21y

2τ

92

+ y2t g

21y

2b

(−

92924− ζ3

)+ y2

t g21g

22

(−

327764

+117

2ζ3

)+ y2

t g41

(−

1231033456

−14912

ζ3 −63572

Ng

)− y2

t λy2τ108

+ y2t λy

2b

(−

3152− 216ζ3

)+ y2

t λg22

(5678− 324ζ3

)+ y2

t λg21

(−

2198− 60ζ3

)+ y2

t λ2 297

2+ y4

t y2τ72 + y4

t y2b

(404716

+ 36ζ3

)+ y4

t g22

(−

317716

+ 162ζ3

)




+ y4t g

21

(−

43116

+ 12ζ3

)+ y4

t λ

(3514

+ 216ζ3

)+ y6

t

(160516

+ 45ζ3

)+ g2

sy4b

(447

2− 360ζ3

)+ g2

sg22y

2b

(−

4894

+ 108ζ3

)+ g2

sg42

(1354

Ng − 36Ngζ3

)+ g2

sg21y

2b

(−

99136

+ 20ζ3

)+ g2

sg41

(554

Ng −443

Ngζ3

)+ g2

sλy2b(−612 + 576ζ3

)+ g2

sy2t y

2b(41 − 48ζ3

)+ g2

sy2t g

22

(−

4894

+ 108ζ3

)+ g2

sy2t g

21

(−

241936

+ 68ζ3

)+ g2

sy2t λ

(−612 + 576ζ3

)+ g2

sy4t

(4472− 360ζ3

)+ g4

sy2b

(9103− 24ζ3 − 32Ng

)+ g4

sy2t

(9103− 24ζ3 − 32Ng

).

FFFFFFFFFF

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Condensed Matter, Particle Physics and Cosmologyfjeger/Krakow_3.pdf · F. Jegerlehner, IFJ-PAN, Krakow Lectures 2014, — nx Lect. 3 xo 147. A summary and outlook of Lecture 2 The