Standard Model Three-loop Beta-functionsGauge couplings.Yukawa couplings. Higgs self-interaction.

Andrey Pikelner

in collaboration with: A.Bednyakov and V.Velizhanin

BLTP,JINR

Erice,2013

Running couplings in the Standard Model

QCD

O(α )

251 MeV

178 MeV

Λ MS(5)

α (Μ )s Z

0.1215

0.1153

0.1

0.2

0.3

0.4

0.5

αs (Q)

1 10 100Q [GeV]

Heavy Quarkonia

Hadron Collisions

e+e- Annihilation

Deep Inelastic Scattering

NL

O

NN

LO

TheoryData

Lat

tice

213 MeV 0.1184s4 {

Figure : Famous running coupling

� Parameters from experiment atscale µ0

� Matching to MS

� Evolution from µ0 to µ usingβαs = dαs(µ)

d log(µ)

� Obtain same at µ

� Compare, tuneparameters,repeat

Scales available at experiment

2 / 15

Running couplings in the Standard Model

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

SMco

uplin

gs

g1

g

gsyt

Λyb

Figure : SM running couplings

� Parameters from experiment atscale µ0 still available

� Matching to MS, precise mt

and mH needed for yt and λ

� Evolution from µ0 to µ usingβi = dgi(µ)

d log(µ)gi = g1, g2, gs, yt, yb, yτ , λ

� Evolution up to scale µ, but allequations are coupledβ = β(g1, g2, gs, yt, yb, yτ , λ)

� For extrapolation we need toextend all β in full model from2-loops to 3-loops

3 / 15

Motivation

1. Is the Standard Model still valid up to a Planck scale?� At the scale Λ ∼ 1018GeV, Λ� µ0 effective potential may be

approximated:

Veff ≈ λ(Λ)Φ4 +O(λ2(Λ), g2i (Λ))

� Vacuum stable if Veff > 0 from µ0 upto Λ� Equal to precise determination of λ(Λ) sign at Λ-scale� Minimal stability bound 129± 3 GeV is very close to mH

2. Calculations in full theory analyticaly w/o additionalassumptions� Computtional methods testing� MS-scheme is used� all fields are massless

4 / 15

Motivation

1. Is the Standard Model still valid up to a Planck scale?� At the scale Λ ∼ 1018GeV, Λ� µ0 effective potential may be

approximated:

Veff ≈ λ(Λ)Φ4 +O(λ2(Λ), g2i (Λ))

� Vacuum stable if Veff > 0 from µ0 upto Λ� Equal to precise determination of λ(Λ) sign at Λ-scale� Minimal stability bound 129± 3 GeV is very close to mH

2. Calculations in full theory analyticaly w/o additionalassumptions� Computtional methods testing� MS-scheme is used� all fields are massless

4 / 15

Available results for beta-functions

� 4-loop QCDT. van Ritbergen, J.A.M. Vermaseren, and S.A. Larin. In: Phys.Lett. B400 (1997)

� 2-loop Standard ModelH. Arason, D.J. Castano, B. Keszthelyi, S. Mikaelian, E.J. Piard, et al. In: Phys.Rev. D46 (1992) Ming-xing Luo

and Yong Xiao. In: Phys.Rev.Lett. 90 (2003) C. Ford, I. Jack, and D.R.T. Jones. In: Nucl.Phys. B387 (1992)

� 3-loop SM gauge couplingsLuminita N. Mihaila, Jens Salomon, and Matthias Steinhauser. In: Phys.Rev.Lett. 108 (2012)

� 3-loop αs + yt + λK.G. Chetyrkin and M.F. Zoller. In: JHEP 1206 (2012)

Full SM results, this workα1, α2, αs A.V. Bednyakov, A.F. Pikelner, and V.N. Velizhanin. In: JHEP 1301 (2013)

yt, yb, yτ A.V. Bednyakov, A.F. Pikelner, and V.N. Velizhanin. In: Phys.Lett. B722 (2013)

λ, µ A.V. Bednyakov, A.F. Pikelner, and V.N. Velizhanin. In: arXiv:1303.4364 [hep-ph] (2013)

5 / 15

Available results for beta-functions

� 4-loop QCDT. van Ritbergen, J.A.M. Vermaseren, and S.A. Larin. In: Phys.Lett. B400 (1997)

� 2-loop Standard ModelH. Arason, D.J. Castano, B. Keszthelyi, S. Mikaelian, E.J. Piard, et al. In: Phys.Rev. D46 (1992) Ming-xing Luo

and Yong Xiao. In: Phys.Rev.Lett. 90 (2003) C. Ford, I. Jack, and D.R.T. Jones. In: Nucl.Phys. B387 (1992)

� 3-loop SM gauge couplingsLuminita N. Mihaila, Jens Salomon, and Matthias Steinhauser. In: Phys.Rev.Lett. 108 (2012)

� 3-loop αs + yt + λK.G. Chetyrkin and M.F. Zoller. In: JHEP 1206 (2012)

Full SM results, this workα1, α2, αs A.V. Bednyakov, A.F. Pikelner, and V.N. Velizhanin. In: JHEP 1301 (2013)

yt, yb, yτ A.V. Bednyakov, A.F. Pikelner, and V.N. Velizhanin. In: Phys.Lett. B722 (2013)

λ, µ A.V. Bednyakov, A.F. Pikelner, and V.N. Velizhanin. In: arXiv:1303.4364 [hep-ph] (2013)

5 / 15

Main ingridients

1. Complicated Feynman rules� Many fields and couplings� A lot of parameters in model

Solution: LanHEP + unbroken model

2. Complicated diagrams� From 10k to 10000k diagrams� Permutations of legs

Solution: FeynArts/Diana(QGRAF)

3. Complicated loop integrals� 2,3,4-point diagrams� Spurious IR-poles

Solution: MINCER+IRR, Massive bubbles

LanHEP

FeynArts

DIANA(QGRAF)

Model

Map

MINCER/MATAD

6 / 15

Main ingridients

1. Complicated Feynman rules� Many fields and couplings� A lot of parameters in model

Solution: LanHEP + unbroken model

2. Complicated diagrams� From 10k to 10000k diagrams� Permutations of legs

Solution: FeynArts/Diana(QGRAF)

3. Complicated loop integrals� 2,3,4-point diagrams� Spurious IR-poles

Solution: MINCER+IRR, Massive bubbles

LanHEP

FeynArts

DIANA(QGRAF)

Model

Map

MINCER/MATAD

6 / 15

Main ingridients

1. Complicated Feynman rules� Many fields and couplings� A lot of parameters in model

Solution: LanHEP + unbroken model

2. Complicated diagrams� From 10k to 10000k diagrams� Permutations of legs

Solution: FeynArts/Diana(QGRAF)

3. Complicated loop integrals� 2,3,4-point diagrams� Spurious IR-poles

Solution: MINCER+IRR, Massive bubbles

LanHEP

FeynArts

DIANA(QGRAF)

Model

Map

MINCER/MATAD

6 / 15

Gauge couplings.Background Field Method

SM in Background Field gauge

LC(V )→ LC(V + V ),√ZViZgi = 1

Vi = (B, W , G), gi = (g1, g2, gs)

Advantages:

� Less gauge fixing parameters

� only 2-point functions needed: MINCER

Disadvantages:

� 2xN fields

� Complicated feynman rules

We implimmented SM in BGF formalism in LanHEP from: Ansgar Denner,

Georg Weiglein, and Stefan Dittmaier. In: Nucl.Phys. B440 (1995)

Our work is generalization for several gauge couplings of this:A.G.M. Pickering, J.A. Gracey, and D.R.T. Jones. In: Phys.Lett. B510 (2001)

7 / 15

Gauge couplings.Results� Unification scale moves to higher energies� Three-loop result is enough

log10

(µ/GeV)

α1, α

2, α

3

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

2 4 6 8 10 12 14 16log10(µ/GeV)

α 1,α2

0.235

0.2353

0.2355

0.2358

0.236

0.2363

0.2365

0.2368

0.237

x10-1

12.98 13 13.02 13.04 13.06 13.08

1-loop

2-loop

3-loop

α1, α2, αsResults for gauge couplings beta-functions, and fieldsanomalous dimensions in computer readable form:

http://arxiv.org/src/1210.6873/anc8 / 15

Yukawa couplings.One more leg

In MS scheme is no dependence on internal masses and distributionof external momentum

Sample Yukawa vertex 3-loop diagram. Higgs leg nullified

→

MINCER topology NO

Problem:

Problem: Naive nullification is dangerous: spurious IR-poles� In general assymptotic expansion in external momentum needed

But: for ffH-vertex it’s not needed

9 / 15

Yukawa couplings. Results

β(3)t ' 1.51y3t − 0.63asy

2t + 0.22a2syt

− 0.11a2y2t

+0.07y2t λ− 0.06a3s

β(3)b ' 1.34y3t − 0.19a2syt − 0.09a3s − 0.06a2y

2t − 0.04a2asyt + 0.03asy

2t

β(3)τ ' 1.19y3t − 0.24asy

2t + 0.09a2syt − 0.04a2y

2t − 0.01y2t λ

Already known result

Full results for Yukawa couplings beta-functions, in computerreadable form: http://arxiv.org/src/1212.6829/anc

10 / 15

Yukawa couplings. Results

β(3)t ' 1.51y3t − 0.63asy

2t + 0.22a2syt − 0.11a2y

2t +0.07y2t λ− 0.06a3s

β(3)b ' 1.34y3t − 0.19a2syt − 0.09a3s − 0.06a2y

2t − 0.04a2asyt + 0.03asy

2t

β(3)τ ' 1.19y3t − 0.24asy

2t + 0.09a2syt − 0.04a2y

2t − 0.01y2t λ

Already known result

Full results for Yukawa couplings beta-functions, in computerreadable form: http://arxiv.org/src/1212.6829/anc

10 / 15

Higgs self-coupling at three loops

Problem:

� Diagrams with four legs

� We cannot safely nullify two of them

Possible solutions:

1. Repeat assymptotic expansion in external momentum twice2. Nullify one momentum, but reduce problem to calculation of

three-loop IR-safe vertex integrals.3. mass as IR-regulator

� Remove all legs

� Artificial mass for all propagators.

� Fully massive bubble integrals are known.

p3

p1

p4

p2p2p5

p5 p4

p6

p2

p1

p3

11 / 15

Higgs self-coupling. Three-loop resultSign of λ - as a test of vacuum stability

100 105 108 1011 1014 1017 1020-0.05

0.00

0.05

0.10

0.15

0.20

Scale Μ, GeV

Λ

Topmass Mt =172.9 GeV

1-loop

2-loop

3-loop

100 105 108 1011 1014 1017 1020-0.05

-0.04

-0.03

-0.02

-0.01

Scale Μ, GeV

Λ

Topmass Mt =172.9 GeV

1-loop

2-loop

3-loop

� Three-loop result is enough for precision

� Main uncertainty from mt and mH

Full results for Higgs self-coupling and mass parameterbeta-functions, in computer readable form:

http://arxiv.org/src/1303.4364/anc12 / 15

Standard Model stability investigation with βλ� λ < 0 if Mh < 111GeV is a strong evidence for new physics, but

excluded in direct measurements� Metastable region coincides with measured mt and mH , λ0

SM asymptotycally free?� Landau-pole is higher than Planck-scale,Λ > MPL,SM is valid

effective theory from Fermi to Planck scales

150 160 170 180 190 200

0.2

0.1

0.0

0.1

Top quarkmass, GeV

Λ

GUT scale Μ 10^18, GeV

130GeV

115GeV

80GeV

Higgs mass

100 105 108 1011 1014 1017 10200.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

Scale Μ, GeV

Λ

Topmass Mt 172.9 GeV

80GeV115GeV126GeV135GeV170GeV

Higgs mass

13 / 15

Strong dependence on SM parameters deviations

100 105 108 1011 1014 1017 1020

0.02

0.00

0.02

0.04

0.06

Scale , GeVμ

Higgs mass Mh 126 GeV

mH+/-2.2mt+/-1.6

as+/-0.0002

mH+/-1.2

λ

dotted:αs ± 0.0002 dashed:mt = 172.9± 1.6GeV ,filled regions:mH = 126± 1.2GeV ,mH ± 2.2GeV

14 / 15

Conclusion

1. Gauge couplings beta-functions in SM are calculated - independentcheck

2. Yukawa couplings beta-functions - new result

3. Higgs self-coupling and mass parameter beta-functions

4. Set of programs for calculations in more complicated models.

5. High precision available in calculations, but comparableprecision in mt and mH determination needed fromexperiment

15 / 15