avoiding death by vacuum decay · w. g. h. avoiding death by vacuum decay. the higgs sector of the...
TRANSCRIPT
![Page 1: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/1.jpg)
Avoiding Death by Vacuum Decay
Cutting on the MSSM parameter space
Wolfgang Gregor Hollik
DESY Hamburg Theory
February 18 2016 | Theory Seminar Zeuthen
W. G. H. Avoiding Death by Vacuum Decay
![Page 2: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/2.jpg)
The end is nigh. . .
W. G. H. Avoiding Death by Vacuum Decay
![Page 3: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/3.jpg)
Motivation and Outline
www.
haar
etz.
com
“I think, we have it!”Rolf Heuer
W. G. H. Avoiding Death by Vacuum Decay
![Page 4: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/4.jpg)
Motivation and Outline
www.
thed
aily
star
.net
“Eureka!”Archimedes
W. G. H. Avoiding Death by Vacuum Decay
![Page 5: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/5.jpg)
What do we have?
[par
ticl
ezoo
]
W. G. H. Avoiding Death by Vacuum Decay
![Page 6: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/6.jpg)
Having pinned down the missing piece. . .
[Hambye, Riesselmann: Phys.Rev. D55 (1997) 7255]
W. G. H. Avoiding Death by Vacuum Decay
![Page 7: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/7.jpg)
The SM phase diagram
0 50 100 150 2000
50
100
150
200
Higgs mass Mh in GeV
Top
mas
sM
tin
GeV
Instability
Non
-perturbativity
Stability
Meta-stab
ility
[Deg
rass
iet
al.JH
EP
1208
(201
2)09
8]
W. G. H. Avoiding Death by Vacuum Decay
![Page 8: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/8.jpg)
Back again
[Hambye, Riesselmann: Phys.Rev. D55 (1997) 7255]
W. G. H. Avoiding Death by Vacuum Decay
![Page 9: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/9.jpg)
Constraining the SM Higgs sector
Theoretical considerationsλ < 4π ↪→ perturbativityλ > 0 ↪→ unbounded from below, aka vacuum stability
trivial at the classical (i.e. tree) level
V (Φ) = m2Φ†Φ + λ(Φ†Φ
)2Quantum level
[courtesy of Max Zoller]
dominant contribution: top quark (yt ∼ 1)dependence on the energy scale: β function!
W. G. H. Avoiding Death by Vacuum Decay
![Page 10: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/10.jpg)
Constraining the SM Higgs sector
Theoretical considerationsλ < 4π ↪→ perturbativityλ > 0 ↪→ unbounded from below, aka vacuum stability
trivial at the classical (i.e. tree) level
V (Φ) = m2Φ†Φ + λ(Φ†Φ
)2Quantum level
[courtesy of Max Zoller]
dominant contribution: top quark (yt ∼ 1)dependence on the energy scale: β function!
W. G. H. Avoiding Death by Vacuum Decay
![Page 11: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/11.jpg)
Energy scale dependence
Scale-independent loop-corrected effective potential
Qd
dQVloop(λi, φ,Q) = 0
Approximation for large field values
Vloop(φ) = λ(φ)φ4,
evaluated at Q ∼ φ
β function for coupling λi
βi(λi) = Qdλi(Q)
dQ
running of λ determines stability of the loop potentialupper bound: Landau pole; lower bound: λ > 0
W. G. H. Avoiding Death by Vacuum Decay
![Page 12: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/12.jpg)
Precise analysis: up to three loops!
[Zoller 2014]
W. G. H. Avoiding Death by Vacuum Decay
![Page 13: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/13.jpg)
Precise analysis: up to three loops!
[Zoller 2014]
W. G. H. Avoiding Death by Vacuum Decay
![Page 14: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/14.jpg)
Stability, instability or metastability?
[Courtesy of Max Zoller]
W. G. H. Avoiding Death by Vacuum Decay
![Page 15: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/15.jpg)
Stability, instability or metastability?
[Courtesy of Max Zoller]
W. G. H. Avoiding Death by Vacuum Decay
![Page 16: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/16.jpg)
Metastability of the Standard Model
mh = 125 GeV: metastable electroweak vacuummetastability: decay time of false vacuum largeinstability scale around 1010...12 GeV
SM sufficiently stableneutrino masses missing
Why New Physics?
Why Supersymmetry?mh = 125 GeV: very suitable for light MSSM Higgsmetastability → absolute stability
Task: Do not introduce further instabilities!(generically difficult)
W. G. H. Avoiding Death by Vacuum Decay
![Page 17: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/17.jpg)
Metastability of the Standard Model
mh = 125 GeV: metastable electroweak vacuummetastability: decay time of false vacuum largeinstability scale around 1010...12 GeV
SM sufficiently stableneutrino masses missing
Why New Physics?
Why Supersymmetry?mh = 125 GeV: very suitable for light MSSM Higgsmetastability → absolute stability
Task: Do not introduce further instabilities!(generically difficult)
W. G. H. Avoiding Death by Vacuum Decay
![Page 18: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/18.jpg)
Metastability of the Standard Model
mh = 125 GeV: metastable electroweak vacuummetastability: decay time of false vacuum largeinstability scale around 1010...12 GeV
SM sufficiently stableneutrino masses missing
Why New Physics?
Why Supersymmetry?mh = 125 GeV: very suitable for light MSSM Higgsmetastability → absolute stability
Task: Do not introduce further instabilities!(generically difficult)
W. G. H. Avoiding Death by Vacuum Decay
![Page 19: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/19.jpg)
Metastability of the Standard Model
mh = 125 GeV: metastable electroweak vacuummetastability: decay time of false vacuum largeinstability scale around 1010...12 GeV
SM sufficiently stableneutrino masses missing
Why New Physics?
Why Supersymmetry?mh = 125 GeV: very suitable for light MSSM Higgsmetastability → absolute stability
Task: Do not introduce further instabilities!
(generically difficult)
W. G. H. Avoiding Death by Vacuum Decay
![Page 20: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/20.jpg)
Metastability of the Standard Model
mh = 125 GeV: metastable electroweak vacuummetastability: decay time of false vacuum largeinstability scale around 1010...12 GeV
SM sufficiently stableneutrino masses missing
Why New Physics?
Why Supersymmetry?mh = 125 GeV: very suitable for light MSSM Higgsmetastability → absolute stability
Task: Do not introduce further instabilities!(generically difficult)
W. G. H. Avoiding Death by Vacuum Decay
![Page 21: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/21.jpg)
Supersymmetry
relates bosonic and fermionic degrees of freedom: absolute zerothe only non-trivial extension of Poincaré symmetry
[Par
ticl
eFe
ver]
WMSSM = µHd·Hu−Y eijHd·LL,iER,j+Y
uijHu·QL,iUR,j−Y d
ijHd·QL,iDR,j
W. G. H. Avoiding Death by Vacuum Decay
![Page 22: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/22.jpg)
Supersymmetry
relates bosonic and fermionic degrees of freedom: absolute zerothe only non-trivial extension of Poincaré symmetry
[Par
ticl
eFe
ver]
WMSSM = µHd·Hu−Y eijHd·LL,iER,j+Y
uijHu·QL,iUR,j−Y d
ijHd·QL,iDR,j
W. G. H. Avoiding Death by Vacuum Decay
![Page 23: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/23.jpg)
Supersymmetry
relates bosonic and fermionic degrees of freedom: absolute zerothe only non-trivial extension of Poincaré symmetry
[Par
ticl
eFe
ver]
WMSSM = µHd·Hu−Y eijHd·LL,iER,j+Y
uijHu·QL,iUR,j−Y d
ijHd·QL,iDR,j
W. G. H. Avoiding Death by Vacuum Decay
![Page 24: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/24.jpg)
The Higgs sector of the MSSM and its stability
WMSSM = µHd·Hu−Y eijHd·LL,iER,j+Y
uijHu·QL,iUR,j−Y d
ijHd·QL,iDR,j
Higgs potential of 2HDM type II
V = m211 H
†dHd +m2
22 H†uHu +
(m2
12 Hu ·Hd + h.c.)
+λ12
(H†dHd
)2+λ22
(H†uHu
)2+ λ3
(H†uHu
)(H†dHd
)+ λ4
(H†uHd
)(H†dHu
)+ {λ5, λ6, λ7}
In the MSSM: tree potential calculated from D-terms and Lsoft
m2 tree
11 = |µ|2 +m2Hd, λtree1 = λtree2 = −λtree3 =
g2 + g′2
4,
m2 tree
22 = |µ|2 +m2Hu, λtree4 =
g2
2,
m2 tree
12 = Bµ, λtree5 = λtree6 = λtree7 = 0.
W. G. H. Avoiding Death by Vacuum Decay
![Page 25: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/25.jpg)
The Higgs sector of the MSSM and its stability
WMSSM = µHd·Hu−Y eijHd·LL,iER,j+Y
uijHu·QL,iUR,j−Y d
ijHd·QL,iDR,j
Higgs potential of 2HDM type II
V = m211 H
†dHd +m2
22 H†uHu +
(m2
12 Hu ·Hd + h.c.)
+λ12
(H†dHd
)2+λ22
(H†uHu
)2+ λ3
(H†uHu
)(H†dHd
)+ λ4
(H†uHd
)(H†dHu
)+ {λ5, λ6, λ7}
In the MSSM: tree potential calculated from D-terms and Lsoft
m2 tree
11 = |µ|2 +m2Hd, λtree1 = λtree2 = −λtree3 =
g2 + g′2
4,
m2 tree
22 = |µ|2 +m2Hu, λtree4 =
g2
2,
m2 tree
12 = Bµ, λtree5 = λtree6 = λtree7 = 0.
W. G. H. Avoiding Death by Vacuum Decay
![Page 26: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/26.jpg)
The Higgs sector of the MSSM and its stability
WMSSM = µHd·Hu−Y eijHd·LL,iER,j+Y
uijHu·QL,iUR,j−Y d
ijHd·QL,iDR,j
Higgs potential of 2HDM type II
V = m211 H
†dHd +m2
22 H†uHu +
(m2
12 Hu ·Hd + h.c.)
+λ12
(H†dHd
)2+λ22
(H†uHu
)2+ λ3
(H†uHu
)(H†dHd
)+ λ4
(H†uHd
)(H†dHu
)+ {λ5, λ6, λ7}
Unbounded from below requirements
λ1 > 0 , λ2 > 0 , λ3 > −√λ1λ2
and others. . . [Gunion, Haber 2003]
always fulfilled in the MSSM @ tree
Extending the treeloop corrections? [Gorbahn, Jäger, Nierste, Trine 2011]
W. G. H. Avoiding Death by Vacuum Decay
![Page 27: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/27.jpg)
Effective 2HDM [Gorbahn, Jäger, Nierste, Trine 2011]
integrating out heavy SUSY particlesrequirement of large SUSY scale MSUSY �MA ∼ veweffective theory: generic 2HDM, λi calculated from SUSYloops
H
H
H
H
tL
tR tL
tR
H
H
H
H
χ0
collecting all SUSY contributions:
λi = λi(tanβ, µ,M1,M2, m2Q, m
2u, m
2d, m
2L, m
2e, Au, Ad, Ae).
W. G. H. Avoiding Death by Vacuum Decay
![Page 28: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/28.jpg)
Effective 2HDM [Gorbahn, Jäger, Nierste, Trine 2011]
integrating out heavy SUSY particlesrequirement of large SUSY scale MSUSY �MA ∼ veweffective theory: generic 2HDM, λi calculated from SUSYloops
H
H
H
H
tL
tR tL
tR
H
H
H
H
χ0
collecting all SUSY contributions:
λi = λi(tanβ, µ,M1,M2, m2Q, m
2u, m
2d, m
2L, m
2e, Au, Ad, Ae).
W. G. H. Avoiding Death by Vacuum Decay
![Page 29: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/29.jpg)
Triviality bounds on the effective 2HDM
simple check:
λ1 > 0 , λ2 > 0 , λ3 > −√λ1λ2,
where now
λi = λtreei +λinoi + λsfermi
16π2.
Severe UFB limitsBounds on λ1,2,3 transfer into bounds on m0, At, µ, . . .
W. G. H. Avoiding Death by Vacuum Decay
![Page 30: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/30.jpg)
Recovery from unbounded from below???
V (φ) = −µ2φ2 + λφ4
W. G. H. Avoiding Death by Vacuum Decay
![Page 31: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/31.jpg)
Recovery from unbounded from below???
V (φ) = −µ2φ2 − λφ4
W. G. H. Avoiding Death by Vacuum Decay
![Page 32: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/32.jpg)
Recovery from unbounded from below???
V (φ) = −µ2φ2 − λφ4+λ(6)φ6
W. G. H. Avoiding Death by Vacuum Decay
![Page 33: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/33.jpg)
Calculating the 1-loop effeective potential
tL
tR
h h+ + + + . . .
+
tL,R
h0u
h0u
+ + . . . +
h0u
h0u
h
h
tL,R
tL,R
tR,L + . . .
dominant contribution from third generation squarksquadrilinear couplings (∼ |Yt|2)trilinear coupling to a linear combination (µ∗Yth
†d −Ath
0u)
series summable to an infinite number of external legs
Do not stop after renormalizable / dim 4 terms!
W. G. H. Avoiding Death by Vacuum Decay
![Page 34: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/34.jpg)
Calculating the 1-loop effeective potential
tL
tR
h h+ + + + . . .
+
tL,R
h0u
h0u
+ + . . . +
h0u
h0u
h
h
tL,R
tL,R
tR,L + . . .
dominant contribution from third generation squarksquadrilinear couplings (∼ |Yt|2)trilinear coupling to a linear combination (µ∗Yth
†d −Ath
0u)
series summable to an infinite number of external legsDo not stop after renormalizable / dim 4 terms!
W. G. H. Avoiding Death by Vacuum Decay
![Page 35: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/35.jpg)
All roads lead to Rome. . .
1-loop effective potential [Coleman, Weinberg 1973]
V1(hu, hd) =1
64π2STrM4(hu, hd)
[ln
(M2(hu, hd)
Q2
)− 3
2
]
field dependent massM(hu, hd)STr accounts for spin degrees of freedom
same result can be obtained by the tadpole method
T ∼ ∂
∂hV1(h) ↪→ V1(h) ∼
∫dh T (h)
[Lee, Sciaccaluga 1975]
functional methods: effective potential for arbitrary number ofscalars: V1(φ1, φ2, . . . φn) [Jackiw 1973]
W. G. H. Avoiding Death by Vacuum Decay
![Page 36: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/36.jpg)
The reasoning behind
The effective potential
Veff(φ): average energy density
The ground state of the theoryVeff minimized:
dVeffdφ
∣∣∣∣φ=v
= 0
Technically:generating function for 1PI n-point Green’s functions:
Veff(φ) = −∞∑n=2
G(n)(pi = 0)φn
conversely: calculate all 1PI n-point functions → Veff(with subleties)
W. G. H. Avoiding Death by Vacuum Decay
![Page 37: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/37.jpg)
The reasoning behind
The effective potential
Veff(φ): average energy density
The ground state of the theoryVeff minimized:
dVeffdφ
∣∣∣∣φ=v
= 0
Technically:generating function for 1PI n-point Green’s functions:
Veff(φ) = −∞∑n=2
G(n)(pi = 0)φn
conversely: calculate all 1PI n-point functions → Veff(with subleties)
W. G. H. Avoiding Death by Vacuum Decay
![Page 38: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/38.jpg)
The reasoning behind
The effective potential
Veff(φ): average energy density
The ground state of the theoryVeff minimized:
dVeffdφ
∣∣∣∣φ=v
= 0
Technically:generating function for 1PI n-point Green’s functions:
Veff(φ) = −∞∑n=2
G(n)(pi = 0)φn
conversely: calculate all 1PI n-point functions → Veff(with subleties)
W. G. H. Avoiding Death by Vacuum Decay
![Page 39: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/39.jpg)
The reasoning behind
The effective potential
Veff(φ): average energy density
The ground state of the theoryVeff minimized:
dVeffdφ
∣∣∣∣φ=v
= 0
Technically:generating function for 1PI n-point Green’s functions:
Veff(φ) = −∞∑n=2
G(n)(pi = 0)φn
conversely: calculate all 1PI n-point functions → Veff
(with subleties)
W. G. H. Avoiding Death by Vacuum Decay
![Page 40: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/40.jpg)
The reasoning behind
The effective potential
Veff(φ): average energy density
The ground state of the theoryVeff minimized:
dVeffdφ
∣∣∣∣φ=v
= 0
Technically:generating function for 1PI n-point Green’s functions:
Veff(φ) = −∞∑n=2
G(n)(pi = 0)φn
conversely: calculate all 1PI n-point functions → Veff(with subleties)
W. G. H. Avoiding Death by Vacuum Decay
![Page 41: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/41.jpg)
Summing up external legs
H
H
H
H
tL
tR tL
tR
+ + + . . .
most dominant contribution from top Yukawa yt and Atcan be easily summed for mtR
= mtL≡M
1-PI potential as generating function for 1-PI Green’s functions
−V1−PI(φ) = Γ1−PI(φ) =∑n
1
n!Gn(pext = 0)φn
“classical” field value φ→ 〈0|φ|0〉dV (φ)dφ = 0 determines ground state of the theory
W. G. H. Avoiding Death by Vacuum Decay
![Page 42: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/42.jpg)
Summing up external legs
H
H
H
H
tL
tR tL
tR
+ + + . . .
h
tL
tR
h = h0†d −Atµ∗Yt
h0u
V1 ∼∑n
ann!2
(h†h)n
,ann!2
=1
n(n− 1)(n− 2)
V1 =NcM
4
32π2[(1 + x)2 log(1 + x) + (1− x)2 log(1− x)− 3x2
]Q2 = M2 x2 = |µYt|2h†h/M4, m2
tL= m2
tR= M2
W. G. H. Avoiding Death by Vacuum Decay
![Page 43: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/43.jpg)
Summing up external legs
H
H
H
H
tL
tR tL
tR
+ + + . . .
h
tL
tR
h = h0†d −Atµ∗Yt
h0u
V1 ∼∑n
ann!2
(h†h)n
,ann!2
=1
n(n− 1)(n− 2)
V1 =NcM
4
32π2[(1 + x)2 log(1 + x) + (1− x)2 log(1− x)− 3x2
]Q2 = M2 x2 = |µYt|2h†h/M4, m2
tL= m2
tR= M2
W. G. H. Avoiding Death by Vacuum Decay
![Page 44: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/44.jpg)
Summing up external legs
H
H
H
H
tL
tR tL
tR
+ + + . . .
h
tL
tR
h = h0†d −Atµ∗Yt
h0u
V1 ∼∑n
ann!2
(h†h)n
,ann!2
=1
n(n− 1)(n− 2)
V1 =NcM
4
32π2[(1 + x)2 log(1 + x) + (1− x)2 log(1− x)− 3x2
]Q2 = M2 x2 = |µYt|2h†h/M4, m2
tL= m2
tR= M2
W. G. H. Avoiding Death by Vacuum Decay
![Page 45: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/45.jpg)
Summing up (ctd.)
Field dependent stop mass
M2t(h0u, h
0d) =
(m2tL
+ |Yth0u|2 Ath0u − µ∗Yth0∗d
A∗th0∗u − µY ∗t h0d m2
tR+ |Yth0u|2
)
trilinear ∼ h(h0d, h0u), quadrilinear ∼ |h0u|2
diagrams with mixed contributions
tL,R
tL,Rh0
u
h0
u
gummi bear factor
(2n+ k − 1)!
k!(2n− 1)! www.idn.uni-bremen.de/biologiedidaktik
W. G. H. Avoiding Death by Vacuum Decay
![Page 46: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/46.jpg)
Summing up (ctd.)
Field dependent stop mass
M2t(h0u, h
0d) =
(m2tL
+ |Yth0u|2 Ath0u − µ∗Yth0∗d
A∗th0∗u − µY ∗t h0d m2
tR+ |Yth0u|2
)
trilinear ∼ h(h0d, h0u), quadrilinear ∼ |h0u|2
diagrams with mixed contributions
h0
u
h0
u
h
h
tL,R
tL,R
tR,L
gummi bear factor
(2n+ k − 1)!
k!(2n− 1)! www.idn.uni-bremen.de/biologiedidaktik
W. G. H. Avoiding Death by Vacuum Decay
![Page 47: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/47.jpg)
Summing up (ctd.)
Field dependent stop mass
M2t(h0u, h
0d) =
(m2tL
+ |Yth0u|2 Ath0u − µ∗Yth0∗d
A∗th0∗u − µY ∗t h0d m2
tR+ |Yth0u|2
)
trilinear ∼ h(h0d, h0u), quadrilinear ∼ |h0u|2
diagrams with mixed contributions
h0
u
h0
u
h
h
tL,R
tL,R
tR,L
gummi bear factor
(2n+ k − 1)!
k!(2n− 1)! www.idn.uni-bremen.de/biologiedidaktik
W. G. H. Avoiding Death by Vacuum Decay
![Page 48: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/48.jpg)
Summing up (ctd.)
Field dependent stop mass
M2t(h0u, h
0d) =
(m2tL
+ |Yth0u|2 Ath0u − µ∗Yth0∗d
A∗th0∗u − µY ∗t h0d m2
tR+ |Yth0u|2
)
trilinear ∼ h(h0d, h0u), quadrilinear ∼ |h0u|2
diagrams with mixed contributions
h0
u
h0
u
h
h
tL,R
tL,R
tR,L
gummi bear factor
(2n+ k − 1)!
k!(2n− 1)! www.idn.uni-bremen.de/biologiedidaktik
W. G. H. Avoiding Death by Vacuum Decay
![Page 49: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/49.jpg)
Summing up (ctd.)
Field dependent stop mass
M2t(h0u, h
0d) =
(m2tL
+ |Yth0u|2 Ath0u − µ∗Yth0∗d
A∗th0∗u − µY ∗t h0d m2
tR+ |Yth0u|2
)
tL
tR
h h+ + + + . . .
+
tL,R
h0
u h0
u
+ + . . . +
h0
u
h0
u
h
h
tL,R
tL,R
tR,L + . . .
W. G. H. Avoiding Death by Vacuum Decay
![Page 50: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/50.jpg)
Summing up (ctd.)
Field dependent stop mass
M2t(h0u, h
0d) =
(m2tL
+ |Yth0u|2 Ath0u − µ∗Yth0∗d
A∗th0∗u − µY ∗t h0d m2
tR+ |Yth0u|2
)
V1 ∼∞∑n=0
∞∑k=0
akn x2nyk , x2 =
|µYt|2h†hM4
, y =|Yth0u|2
M2
=
∞∑n=0
∞∑k=0
1
n(2n+ k − 1)(2n+ k − 2)
(2n+ k − 1)!
k!(2n− 1)!x2nyk
=[
(1 + y + x)2 log(1 + y + x)
+(1 + y − x)2 log(1 + y − x)− 3(x2 + y2 + 2y)]
W. G. H. Avoiding Death by Vacuum Decay
![Page 51: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/51.jpg)
Features of the resummed series
V1(h0u, h
0d) =
NcM4
32π2
[(1 + y + x)2 log(1 + y + x)
+ (1 + y − x)2 log(1 + y − x)
− 3(x2 + y2 + 2y)
]
x2 =|µYt|2h†hM4
, h = h0∗d −Atµ∗Yt
h0u, y =|Yth0u|2
M2
branch cut at x− y = ±1: take real part (analytic cont.)ignore imaginary part: log(1 + y − x) = 1
2 log((1 + y − x)2
)always bounded from belowminimum independent of Higgs parameters from tree potentialminimum determined by SUSY scale parameters
W. G. H. Avoiding Death by Vacuum Decay
![Page 52: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/52.jpg)
Tree, loop and tree + loop
(a)
x
V(x)
10.90.80.70.60.50.40.30.20.10
0.5
0.4
0.3
0.2
0.1
0
-0.1
W. G. H. Avoiding Death by Vacuum Decay
![Page 53: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/53.jpg)
Tree, loop and tree + loop
(b)
x
V(x)
43.532.521.510.50
8
6
4
2
0
-2
-4
W. G. H. Avoiding Death by Vacuum Decay
![Page 54: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/54.jpg)
Tree, loop and tree + loop
(c)
x
V(x)
1.41.210.80.60.40.20
4
3
2
1
0
-1
-2
-3
W. G. H. Avoiding Death by Vacuum Decay
![Page 55: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/55.jpg)
Tree, loop and tree + loop
(d)
x
V(x)
1.41.210.80.60.40.20
0.15
0.1
0.05
0
-0.05
-0.1
-0.15
W. G. H. Avoiding Death by Vacuum Decay
![Page 56: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/56.jpg)
Cooking up phenomenologically viable parameters
Minimum at the electroweak scale v = 246 GeV
m2 tree
11 = m2 tree
12 tanβ − v2
2cos(2β)λtree1 − 1
v cosβ
δ
δφdV1
∣∣∣∣φu,d→ 0χu,d→ 0
,
m2 tree
22 = m2 tree
12 cotβ +v2
2cos(2β)λtree1 − 1
v sinβ
δ
δφuV1
∣∣∣∣φu,d→ 0χu,d→ 0
.
mh = 125 GeV
using FeynHiggs 2.10.0 to determine light Higgs mass byadjusting At (several solutions: signAt = − signµ)connection to potential: mA
pseudoscalar mass mA less dependent on higher loopsdecoupling limit: mA,mH± ,mH � mh
include sbottom (drives minimum), take Ab = 0
W. G. H. Avoiding Death by Vacuum Decay
![Page 57: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/57.jpg)
tanβ resummation for bottom yukawa coupling
Yukawa coupling not given directly by the mass
yb =mb
vd(1 + ∆b)
∆gluinob =
2αs3π
µMG tanβC0(mb1, mb2
,MG),
∆higgsinob =
Y 2t
16π2µAt tanβC0(mt1
, mt2, µ).
20 000 40 000 60 000 80 000 100 000
MG
GeV
0.4
0.6
0.8
1.0
1.2
DbHgluinoL
-4000 -2000 2000 4000Μ
-0.2
-0.1
0.1
0.2
DbHhiggsinoL
W. G. H. Avoiding Death by Vacuum Decay
![Page 58: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/58.jpg)
Discussion of stability
✷ ✂ ✶✵✽
✹ ✂ ✶✵✽
✻ ✂ ✶✵✽
�✷ ✂ ✶✵✽
✷✵✵ ✹✵✵ ✻✵✵
✈✉ ✰ ✣✉ ❂●❡❱
✁✄❢❢ ❂ ●❡❱☎
✵
☎
✽
✆✝
tanβ = 40
mA = 800 GeV
M = 1 TeV
µ = 3.75 TeV
At ' −1.5 TeV
W. G. H. Avoiding Death by Vacuum Decay
![Page 59: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/59.jpg)
Discussion of stability
✷ ✂ ✶✵✽
✹ ✂ ✶✵✽
✻ ✂ ✶✵✽
�✷ ✂ ✶✵✽
✷✵✵ ✹✵✵ ✻✵✵
✈✉ ✰ ✣✉ ❂●❡❱
✁✄❢❢ ❂ ●❡❱☎
✵
☎✽
✆✝
tanβ = 40
mA = 800 GeV
M = 1 TeV
µ = 3.83 TeV
At ' −1.5 TeV
W. G. H. Avoiding Death by Vacuum Decay
![Page 60: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/60.jpg)
[ww
w.b
bcam
eric
a.co
m]
W. G. H. Avoiding Death by Vacuum Decay
![Page 61: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/61.jpg)
Naive exclusions: Constraint in µ-tanβ
MSUSY = 2TeVMSUSY = 1TeV
µ/TeV
tanβ
54321
60
50
40
30
20
10
W. G. H. Avoiding Death by Vacuum Decay
![Page 62: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/62.jpg)
New interpretation
Access to Charge and Color breaking minima
µ = 3910 GeVµ = 3860 GeVµ = 3810 GeV
vu +ϕu/GeV
V eff/G
eV4
6004002000
6× 108
5× 108
4× 108
3× 108
2× 108
1× 108
0× 1
−1× 108
−2× 108
−3× 108
−4× 108
W. G. H. Avoiding Death by Vacuum Decay
![Page 63: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/63.jpg)
New interpretation
Access to Charge and Color breaking minima
M2t(h0u, h
0d) =
(m2Q + |Yth0u|2 Ath
0u − µ∗Yth0∗d
A∗th0∗u − µY ∗t h0d m2
t + |Yth0u|2)
M2b(h0u, h
0d) =
(m2Q + |Ybh0d|2 Abh
0d − µ∗Ybh0∗u
A∗bh0∗d − µY ∗b h0u m2
b + |Ybh0d|2)
non-trivial behaviour of sfermions masses with Higgs vev
:
m2b1,2
(h0u, h0d) =
m2Q + m2
b
2+ |Ybh0d|2
± 1
2
√(m2
Q − m2b)
2 + 4|Abh0d − µ∗Ybh0∗u |2
expand theory around new minimum: m2b2< 0
tachyonic squark mass!
W. G. H. Avoiding Death by Vacuum Decay
![Page 64: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/64.jpg)
New interpretation
Access to Charge and Color breaking minima
M2t(h0u, h
0d) =
(m2Q + |Yth0u|2 Ath
0u − µ∗Yth0∗d
A∗th0∗u − µY ∗t h0d m2
t + |Yth0u|2)
M2b(h0u, h
0d) =
(m2Q + |Ybh0d|2 Abh
0d − µ∗Ybh0∗u
A∗bh0∗d − µY ∗b h0u m2
b + |Ybh0d|2)
non-trivial behaviour of sfermions masses with Higgs vev:
m2b1,2
(h0u, h0d) =
m2Q + m2
b
2+ |Ybh0d|2
± 1
2
√(m2
Q − m2b)
2 + 4|Abh0d − µ∗Ybh0∗u |2
expand theory around new minimum: m2b2< 0
tachyonic squark mass!
W. G. H. Avoiding Death by Vacuum Decay
![Page 65: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/65.jpg)
New interpretation
Access to Charge and Color breaking minima
M2t(h0u, h
0d) =
(m2Q + |Yth0u|2 Ath
0u − µ∗Yth0∗d
A∗th0∗u − µY ∗t h0d m2
t + |Yth0u|2)
M2b(h0u, h
0d) =
(m2Q + |Ybh0d|2 Abh
0d − µ∗Ybh0∗u
A∗bh0∗d − µY ∗b h0u m2
b + |Ybh0d|2)
non-trivial behaviour of sfermions masses with Higgs vev:
m2b1,2
(h0u, h0d) =
m2Q + m2
b
2+ |Ybh0d|2
± 1
2
√(m2
Q − m2b)
2 + 4|Abh0d − µ∗Ybh0∗u |2
expand theory around new minimum: m2b2< 0
tachyonic squark mass!
W. G. H. Avoiding Death by Vacuum Decay
![Page 66: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/66.jpg)
[com
mons
.wik
imed
ia.o
rg]
W. G. H. Avoiding Death by Vacuum Decay
![Page 67: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/67.jpg)
Rolling down the hill
What does a tachyonic mass mean?
mass ⇔ second derivative: m2φ = ∂2V/∂φ2
m2φ < 0 ⇔ negative curvature
non-convex potential: imaginary partlog(1 + y − x) ∼ log(m2
φ)
-100 -50 50 100 b�
L
-2´107
2´107
4´107
6´107
8´107
1´108
V Hb� LL
W. G. H. Avoiding Death by Vacuum Decay
![Page 68: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/68.jpg)
Rolling down the hill
What does a tachyonic mass mean?
mass ⇔ second derivative: m2φ = ∂2V/∂φ2
m2φ < 0 ⇔ negative curvature
non-convex potential: imaginary partlog(1 + y − x) ∼ log(m2
φ)
-100 -50 50 100 b�
L
-2´107
2´107
4´107
6´107
8´107
1´108
V Hb� LL
W. G. H. Avoiding Death by Vacuum Decay
![Page 69: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/69.jpg)
A more complete picture
Including colored directions
V treeb
= b∗L(M2Q
+ |Ybvd|2)bL + b∗R(M2b
+ |Ybvd|2)bR
−[b∗L(µ∗Ybh
0†u −Abvd)bR + h. c.
]+ |Yb|2|bL|2|bR|2
+ D-terms.
D-flat direction: minimizing D-term contribution
D-terms: g2φ4
VD =g218
(|h0u|2 − |h0d|2 +
1
3|bL|2 +
2
3|bR|2
)2+g228
(|h0u|2 − |h0d|2 + |bL|2
)2+g236
(|bL|2 − |bR|2
)2.
will always take over for large field valuestake e.g. bL = bR = b and |h0d|2 = |h0u|2 + |b|2|
W. G. H. Avoiding Death by Vacuum Decay
![Page 70: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/70.jpg)
A more complete picture
Including colored directions
V treeb
= b∗L(M2Q
+ |Ybvd|2)bL + b∗R(M2b
+ |Ybvd|2)bR
−[b∗L(µ∗Ybh
0†u −Abvd)bR + h. c.
]+ |Yb|2|bL|2|bR|2
+ D-terms.
D-flat direction: minimizing D-term contribution
D-terms: g2φ4
VD =g218
(|h0u|2 − |h0d|2 +
1
3|bL|2 +
2
3|bR|2
)2+g228
(|h0u|2 − |h0d|2 + |bL|2
)2+g236
(|bL|2 − |bR|2
)2.
will always take over for large field valuestake e.g. bL = bR = b and |h0d|2 = |h0u|2 + |b|2|
W. G. H. Avoiding Death by Vacuum Decay
![Page 71: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/71.jpg)
A more complete picture
Including colored directions
V treeb
= b∗L(M2Q
+ |Ybvd|2)bL + b∗R(M2b
+ |Ybvd|2)bR
−[b∗L(µ∗Ybh
0†u −Abvd)bR + h. c.
]+ |Yb|2|bL|2|bR|2
+ D-terms.
D-flat direction: minimizing D-term contribution
D-terms: g2φ4
VD =g218
(|h0u|2 − |h0d|2 +
1
3|bL|2 +
2
3|bR|2
)2+g228
(|h0u|2 − |h0d|2 + |bL|2
)2+g236
(|bL|2 − |bR|2
)2.
will always take over for large field valuesor h0d = 0 and b = h0u [large D-term]
W. G. H. Avoiding Death by Vacuum Decay
![Page 72: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/72.jpg)
CCB minima and the one-loop Higgs potential
From CCC to CCBpreviously “safe” false but CC conserving minima turn into todeep global minima with 〈bL〉 = 〈bR〉 6= 0 and 〈h0u〉 6= vu
W. G. H. Avoiding Death by Vacuum Decay
![Page 73: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/73.jpg)
CCB minima and the one-loop Higgs potential
From CCC to CCBpreviously “safe” false but CC conserving minima turn into todeep global minima with 〈bL〉 = 〈bR〉 6= 0 and 〈h0u〉 6= vu
W. G. H. Avoiding Death by Vacuum Decay
![Page 74: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/74.jpg)
Analytic bounds at the tree-level
1 choose appropriate direction ↪→ one-field problem
V treeφ = m2φ2 −Aφ3 + λφ4
h0u = b, h0d = 0
2 identify parameters, e.g. m2 = m2Q + m2
b +m2Hu
+ µ2,
λ = Y 2b +
g21+g22
2 , A = 2µYb
3 necessary condition: m2 > A2
4λ ↪→ second minimum not belowfirst (trivial) one)
h0u = b, h0
d = 0
m2Hu
+ µ2 + m2Q + m2
b >(µYb)2
Y 2b + (g21 + g22)/2
W. G. H. Avoiding Death by Vacuum Decay
![Page 75: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/75.jpg)
Analytic bounds at the tree-level
1 choose appropriate direction ↪→ one-field problem
V treeφ = m2φ2 −Aφ3 + λφ4
h0u = b, h0d = 0
2 identify parameters, e.g. m2 = m2Q + m2
b +m2Hu
+ µ2,
λ = Y 2b +
g21+g22
2 , A = 2µYb
3 necessary condition: m2 > A2
4λ ↪→ second minimum not belowfirst (trivial) one)
h0u = b, h0
d = 0
m2Hu
+ µ2 + m2Q + m2
b >(µYb)2
Y 2b + (g21 + g22)/2
W. G. H. Avoiding Death by Vacuum Decay
![Page 76: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/76.jpg)
Analytic bounds at the tree-level
1 choose appropriate direction ↪→ one-field problem
V treeφ = m2φ2 −Aφ3 + λφ4
h0u = b, h0d = 0
2 identify parameters, e.g. m2 = m2Q + m2
b +m2Hu
+ µ2,
λ = Y 2b +
g21+g22
2 , A = 2µYb
3 necessary condition: m2 > A2
4λ ↪→ second minimum not belowfirst (trivial) one)
|h0d|2 = |h0
u|2 + |b|2, b = αh0u
m211(1 + α2) +m2
22 ± 2m212
√1 + α2 + α2(m2
Q + m2b) >
µ2α4
2 + 3α2
W. G. H. Avoiding Death by Vacuum Decay
![Page 77: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/77.jpg)
µ/GeV
tanβ
5000450040003500300025002000
60
50
40
30
20
10
0
W. G. H. Avoiding Death by Vacuum Decay
![Page 78: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/78.jpg)
Conclusions
the Higgs potential in the SM is meta/un/stableMSSM: multi-scalar theory, has several unwanted minimaformation of new CCB conserving minima at the 1-loop levelstability of the electroweak vacuum: bounds on µ-tanβ
instability of electroweak vacuum by second minimum in“standard model direction” ∼ vu: global CCB minimummore severe bounds [new CCB constraints]CCB constraints for non-vanishing D-Termsquantum tunneling: either very long- or very short-lived
W. G. H. Avoiding Death by Vacuum Decay
![Page 79: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/79.jpg)
Finally. . .
Greetings from Señor Higgs(courtesy of Jens Hoff)
W. G. H. Avoiding Death by Vacuum Decay
![Page 80: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/80.jpg)
Backup
Slides
W. G. H. Avoiding Death by Vacuum Decay
![Page 81: Avoiding Death by Vacuum Decay · W. G. H. Avoiding Death by Vacuum Decay. The Higgs sector of the MSSM and its stability W MSSM = H dH u Y e ij H dL L;iE R;j+Y u ij H uQ L;iU R;j](https://reader035.vdocument.in/reader035/viewer/2022071010/5fc7841ecb19f70c654411a7/html5/thumbnails/81.jpg)
References
Wolfgang Gregor Hollik: “Charge and color breaking constraints inthe Minimal Supersymmetric Standard Model associated with thebottom Yukawa coupling” Physics Letters B 752 (2016) 7 – 12
Wolfgang Gregor Hollik: “Neutrinos meet Supersymmetry: QuantumAspects of Neutrinophysics in Supersymmetric Theories”, PhDThesis, Karlsruhe 2015
Markus Bobrowski, Guillaume Chalons, Wolfgang G. Hollik, UlrichNierste: “Vacuum stability of the effective Higgs potential in theMinimal Supersymmetric Standard Model” Physical Review D 90,035025 (2014)
W. G. H. Avoiding Death by Vacuum Decay