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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