Higgs and Dark Matter Hints of an Oasis in the...
Transcript of Higgs and Dark Matter Hints of an Oasis in the...
Higgs and Dark Matter Hintsof an Oasis in the Desert
Clifford Cheung
University of California, BerkeleyLawrence Berkeley National Lab
Michigan Higgs SymposiumC.C., Papucci, Zurek (1203.5106)
Combined exclusion limit
Zoom in:
Expected exclusion at 95% CL: 120-555 GeV
Observed exclusion at 95% CL: 110-117.5, 118.5-122.5, 129-539 GeV
Observed exclusion at 99% CL: 130-486 GeV
Introduction / High-mH search: νν, jj, νjj / Low-mH search: 4, γγ • νν, bb, ττ / Combination / End? 21/24
Combined exclusion limit
Zoom in:
Expected exclusion at 95% CL: 120-555 GeV
Observed exclusion at 95% CL: 110-117.5, 118.5-122.5, 129-539 GeV
Observed exclusion at 99% CL: 130-486 GeV
Introduction / High-mH search: νν, jj, νjj / Low-mH search: 4, γγ • νν, bb, ττ / Combination / End? 21/24
I’m going to take this seriously.
• running:
What are the implications of mh = 125 GeV?
• naturalness:
strong implications
dλH
d logQ= βλH
exponentially sensitive to mh
• running:
What are the implications of mh = 125 GeV?
• naturalness:
strong implications
exponentially sensitive to mh
dλH
d logQ= βλH
At loop level, the quartic runs with scale.
Running to high energies, the quartic may approach one of two special values.
4π0
strongcoupling
unstablevacuum
λ(µ)
(4π)2dλ
d logQ= 12λ2
H− 12y4
t+ 12λy2
t+ . . .
176 J.A. Casas et al. / Physics Letters B 342 (1995) 171-179
0.04
0.03
0.02 ,<
z " < 0.01
0 .00
- 0 . 0 1
75
25
~ o
~ . -25
• ~ -50
, \ . . . . . . . . . . . . 1~ A = 10 Is OeV \ ~ as = 0.124
M. = 100 GeV
Mt = 160 GeV
10 16 18 - 0 . 0 2 ' ' ' ' ' ' ' ' ' ' ' ' ' J ' ' ' ' - 7 5
0 2 4 6 8 12 14 0
Iog,o[/~/GeV]
M. = 100 OeV ] Id, = 160 GeV
, 1 , I t i i I , I t I t I , I , t
2 4 6 8 10 12 14 16 18
log,o[~/GeV]
Fig. 2. (a) Plot of A (dashed line) and A. (solid line) as a function of the scale/z(t) for A and as as in Fig. 1, Mt = 160 GeV and MH = 100 GeV. (b) Plot of the scalar potential, V(~b), corresponding to Fig. 2a, represented in a convenient choice of units as described in the text.
0.15 i i i i i i [ , i i i , i , i
0.10
oo 0.00
-0.05 h = 10 is as = 0.124 )4. = 85 OeV )4~ = 160 GeV
- - 0 . 1 0 , i , t , I , ' i , i , i , i
2 4 6 8 10 12 14 16
Iog,oE/z,/Idz]
Fig. 3. Plot of A as a function of the scale/z(t) for Mt = 160 GeV, A = 1019 GeV and as = 0.124. The curves are shown for intervals of 5 GeV in MH in the range 85 GeV < MH < 115 GeV. The thicker (tangent) line corresponds to the case MH = 101 GeV.
( M e = 101 GeV) , which is tangent to the horizontal axis at A0 = 4 x 1011 GeV, represents a l imiting case: for MH > 101 GeV the potential has no maximum and therefore is completely safe for any choice of A [see condition ( a ) above] ; for Mn < 101 GeV the models are safe depending on the value of A < A0. Hence, for A > A0 the lower bound on MH is insensitive to the
value of A. The situation depicted in this example, i.e. for Mt = 160 GeV, typically occurs when the top mass is rather small, since then the top Yukawa coupling is too small at large scales to maintain fla negative [ see Eq. (10) ]. When the top mass is higher the situation is different and is illustrated with the case M t = 174 GeV in Fig. 4. There we see that for each value of MH, ~ ( t ) crosses the horizontal axis at most once, and there is a value of Mn for which the cross occurs at A = 1019 GeV. In consequence, for A < 1019 GeV there is a one-to-one correspondence between the choice of A and the value of the lower bound on MH.
Finally, we would like to point out that generically the potential is not unbounded from below since when- ever there is a maximum, there is an additional mini- mum for a larger value of ~b, as illustrated in Fig. 2b. This is because ~ gets back to the posit iye rage for large enough scales. Hence, the potential becomes eventually positive and monotonical ly increasing, al- though in some cases, e.g. for Mt = 174 GeV, this occurs for values of ~ beyond 1019 GeV.
5. Numerical results and comparison with SUSY bounds
As stated in the Introduction and has become clear in previous sections, the lower bound on M e is a func-
Veff(φ)φ→∞=
λ(µ = |φ|)2
|φ|4
Phys.Lett. B342 (1995) 171-179
176 J.A. Casas et al. / Physics Letters B 342 (1995) 171-179
0.04
0.03
0.02 ,<
z " < 0.01
0 .00
- 0 . 0 1
75
25
~ o
~ . -25
• ~ -50
, \ . . . . . . . . . . . . 1~ A = 10 Is OeV \ ~ as = 0.124
M. = 100 GeV
Mt = 160 GeV
10 16 18 - 0 . 0 2 ' ' ' ' ' ' ' ' ' ' ' ' ' J ' ' ' ' - 7 5
0 2 4 6 8 12 14 0
Iog,o[/~/GeV]
M. = 100 OeV ] Id, = 160 GeV
, 1 , I t i i I , I t I t I , I , t
2 4 6 8 10 12 14 16 18
log,o[~/GeV]
Fig. 2. (a) Plot of A (dashed line) and A. (solid line) as a function of the scale/z(t) for A and as as in Fig. 1, Mt = 160 GeV and MH = 100 GeV. (b) Plot of the scalar potential, V(~b), corresponding to Fig. 2a, represented in a convenient choice of units as described in the text.
0.15 i i i i i i [ , i i i , i , i
0.10
oo 0.00
-0.05 h = 10 is as = 0.124 )4. = 85 OeV )4~ = 160 GeV
- - 0 . 1 0 , i , t , I , ' i , i , i , i
2 4 6 8 10 12 14 16
Iog,oE/z,/Idz]
Fig. 3. Plot of A as a function of the scale/z(t) for Mt = 160 GeV, A = 1019 GeV and as = 0.124. The curves are shown for intervals of 5 GeV in MH in the range 85 GeV < MH < 115 GeV. The thicker (tangent) line corresponds to the case MH = 101 GeV.
( M e = 101 GeV) , which is tangent to the horizontal axis at A0 = 4 x 1011 GeV, represents a l imiting case: for MH > 101 GeV the potential has no maximum and therefore is completely safe for any choice of A [see condition ( a ) above] ; for Mn < 101 GeV the models are safe depending on the value of A < A0. Hence, for A > A0 the lower bound on MH is insensitive to the
value of A. The situation depicted in this example, i.e. for Mt = 160 GeV, typically occurs when the top mass is rather small, since then the top Yukawa coupling is too small at large scales to maintain fla negative [ see Eq. (10) ]. When the top mass is higher the situation is different and is illustrated with the case M t = 174 GeV in Fig. 4. There we see that for each value of MH, ~ ( t ) crosses the horizontal axis at most once, and there is a value of Mn for which the cross occurs at A = 1019 GeV. In consequence, for A < 1019 GeV there is a one-to-one correspondence between the choice of A and the value of the lower bound on MH.
Finally, we would like to point out that generically the potential is not unbounded from below since when- ever there is a maximum, there is an additional mini- mum for a larger value of ~b, as illustrated in Fig. 2b. This is because ~ gets back to the posit iye rage for large enough scales. Hence, the potential becomes eventually positive and monotonical ly increasing, al- though in some cases, e.g. for Mt = 174 GeV, this occurs for values of ~ beyond 1019 GeV.
5. Numerical results and comparison with SUSY bounds
As stated in the Introduction and has become clear in previous sections, the lower bound on M e is a func-
Veff(φ)φ→∞=
λ(µ = |φ|)2
|φ|4
Phys.Lett. B342 (1995) 171-179
vacuum decay
vacuum structure
Depending on the quartic, the vacuum is:
i) stable :
ii) metastable :
iii) unstable :
where depends on age of the universe.
λH > 0
0 > λH > λH
λH > λH
λH
vacuum structure
Depending on the quartic, the vacuum is:
i) stable :
ii) metastable :
iii) unstable :
where depends on age of the universe.
λH > 0
0 > λH > λH
λH > λH
λH
excluded by our existence
λH = λH
λH = 0
stable
metastable
unstable
104 106 108 1010 1012 1014 1016 10180.15
0.10
0.05
0.00
0.05
0.10
GeV
Λ
λH = λH
λH = 0
stable
metastable
unstable
104 106 108 1010 1012 1014 1016 10180.15
0.10
0.05
0.00
0.05
0.10
GeV
Λ
oasis required
oasis required if new states restore SUSY
stable
metastable
unstable
106 107 108 109 1010 1011 1012 1013 1014 1015
100
110
120
130
GeV
mHGeV
SM
Figure 1: Vacuum structure of the SM as a function of the Higgs boson mass. Regions of
stability/metastability/instability are denoted in blue/purple/red respectively. The solid lines
indicate central values while the dotted lines indicate ±2σ error bars on the experimental mea-
surement of the top quark mass.
ii) Metastable (0 > λH > λH). The vacuum is not the absolute minimum, but its lifetime is
longer than the age of the Universe.
iii) Unstable (λH > λH). The vacuum is not the absolute minimum, and it decays within the
age of the Universe.
Here the critical coupling λH is determined by the requirement that the tunneling rate per unit
volume is comparable to the age of the Universe. In particular, we demand that H4= Γ, where
H−1 3.7 Gyr and Γ reads,
Γ = max
R
−4exp(−16π2
/3|λH |)
R−1<Λ
. (1)
Here R is the characteristic length scale of the bounce, which is bounded by the cutoff. As
we will elaborate on later, the vacuum structure may be more complicated if the new physics
includes additional scalar particles.
It is possible that our vacuum resides in a stable or metastable regime, but the unstable
regime is of course excluded by our existence. In much of our analyses, it will be convenient to
3
• flavor puzzle
But physics beyond the SM is required!
• hierarchy problem
• strong CP
• baryon asymmetry
• neutrino masses
• dark matter
• flavor puzzle
• hierarchy problem
• strong CP
• baryon asymmetry
• neutrino masses
• dark matter
tuningdynamics
But physics beyond the SM is required!
• flavor puzzle
• hierarchy problem
• strong CP
• baryon asymmetry
• neutrino masses
• dark matter
tuningdynamics
But physics beyond the SM is required!
• flavor puzzle
• hierarchy problem
• strong CP
• baryon asymmetry
• neutrino masses
• dark matter
tuningdynamics
But physics beyond the SM is required!
high scalephysics?
WIMP miracle
The present day abundance of dark matter,
is more or less correct given a weak-scale annihilation cross-section.
Ωh2 1 pb
σv
question of this talk
What is the status of the “desert” in light of the SM plus thermal relic DM?
• cutoff from non-perturbativity?
• cutoff from instability?
• all subject to Ωh2=0.11 and σSI.
models+
methodology
fermionic DM
0.2
0.3
0.1
0.
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115
120
125
130
GeV
mHGeV
1.62.
1.2
0.8
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120
140
160
180
200
220
240
GeV
mHGeV
SM Triplet Scalar
Figure 4: Same as Fig. (3) but for SM + triplet scalar and varying κT with λT=0.
For the case of new fermions we consider the following theories:
singlet/doublet fermion: −∆L =1
2mSS
2+mDDD
c+ ySHSD + y
cSH
cSD
c
triplet/doublet fermion: −∆L =1
2mTT
2+mDDD
c+ yTHTD + y
cTH
cTD
c,
(4)
where we have defined Hc ≡ H∗
, and we have included all renormalizable operators permitted
by gauge symmetry. Here S and T are Majorana fermions and D and Dcform a Dirac fermion.
As a consequence of the additional Yukawa couplings to the Higgs, the new fermionic states will
generally mix after electroweak symmetry breaking, although there is a preserved Z2 symmetry
which makes the lightest among these fields a DM candidate. We do not consider the fully
mixed singlet/triplet/doublet theory, since all the qualitative features of this model are already
evident in the singlet/doublet and triplet/doublet cases, while the proliferation of parameters
would make results difficult to present.
The fermionic theories above are of course a generalization of the bino, wino, and Higgsino
sector of the MSSM. In the chosen parameterization, the precise correspondence between these
6
0.2
0.3
0.1
0.
106 108 1010 1012 1014 1016
115
120
125
130
GeV
mHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Triplet Scalar
Figure 4: Same as Fig. (3) but for SM + triplet scalar and varying κT with λT=0.
For the case of new fermions we consider the following theories:
singlet/doublet fermion: −∆L =1
2mSS
2+mDDD
c+ ySHSD + y
cSH
cSD
c
triplet/doublet fermion: −∆L =1
2mTT
2+mDDD
c+ yTHTD + y
cTH
cTD
c,
(4)
where we have defined Hc ≡ H∗
, and we have included all renormalizable operators permitted
by gauge symmetry. Here S and T are Majorana fermions and D and Dcform a Dirac fermion.
As a consequence of the additional Yukawa couplings to the Higgs, the new fermionic states will
generally mix after electroweak symmetry breaking, although there is a preserved Z2 symmetry
which makes the lightest among these fields a DM candidate. We do not consider the fully
mixed singlet/triplet/doublet theory, since all the qualitative features of this model are already
evident in the singlet/doublet and triplet/doublet cases, while the proliferation of parameters
would make results difficult to present.
The fermionic theories above are of course a generalization of the bino, wino, and Higgsino
sector of the MSSM. In the chosen parameterization, the precise correspondence between these
6
• singlet/doublet
• triplet/doublet
(Hc ≡ H∗)
fermionic DM
0.2
0.3
0.1
0.
106 108 1010 1012 1014 1016
115
120
125
130
GeV
mHGeV
1.62.
1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Triplet Scalar
Figure 4: Same as Fig. (3) but for SM + triplet scalar and varying κT with λT=0.
For the case of new fermions we consider the following theories:
singlet/doublet fermion: −∆L =1
2mSS
2+mDDD
c+ ySHSD + y
cSH
cSD
c
triplet/doublet fermion: −∆L =1
2mTT
2+mDDD
c+ yTHTD + y
cTH
cTD
c,
(4)
where we have defined Hc ≡ H∗
, and we have included all renormalizable operators permitted
by gauge symmetry. Here S and T are Majorana fermions and D and Dcform a Dirac fermion.
As a consequence of the additional Yukawa couplings to the Higgs, the new fermionic states will
generally mix after electroweak symmetry breaking, although there is a preserved Z2 symmetry
which makes the lightest among these fields a DM candidate. We do not consider the fully
mixed singlet/triplet/doublet theory, since all the qualitative features of this model are already
evident in the singlet/doublet and triplet/doublet cases, while the proliferation of parameters
would make results difficult to present.
The fermionic theories above are of course a generalization of the bino, wino, and Higgsino
sector of the MSSM. In the chosen parameterization, the precise correspondence between these
6
0.2
0.3
0.1
0.
106 108 1010 1012 1014 1016
115
120
125
130
GeV
mHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Triplet Scalar
Figure 4: Same as Fig. (3) but for SM + triplet scalar and varying κT with λT=0.
For the case of new fermions we consider the following theories:
singlet/doublet fermion: −∆L =1
2mSS
2+mDDD
c+ ySHSD + y
cSH
cSD
c
triplet/doublet fermion: −∆L =1
2mTT
2+mDDD
c+ yTHTD + y
cTH
cTD
c,
(4)
where we have defined Hc ≡ H∗
, and we have included all renormalizable operators permitted
by gauge symmetry. Here S and T are Majorana fermions and D and Dcform a Dirac fermion.
As a consequence of the additional Yukawa couplings to the Higgs, the new fermionic states will
generally mix after electroweak symmetry breaking, although there is a preserved Z2 symmetry
which makes the lightest among these fields a DM candidate. We do not consider the fully
mixed singlet/triplet/doublet theory, since all the qualitative features of this model are already
evident in the singlet/doublet and triplet/doublet cases, while the proliferation of parameters
would make results difficult to present.
The fermionic theories above are of course a generalization of the bino, wino, and Higgsino
sector of the MSSM. In the chosen parameterization, the precise correspondence between these
6
• singlet/doublet
• triplet/doublet
(Hc ≡ H∗)vector-likefor anomalies
required for singlet-like thermal DM
fermionic DM
0.2
0.3
0.4
0.10.
106 108 1010 1012 1014 1016
115
120
125
130
GeVmHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
0.2
0.3
0.4
0.10.
106 108 1010 1012 1014 1016
115
120
125
130
GeV
mHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
0.2
0.3
0.4
0.10.
106 108 1010 1012 1014 1016
115
120
125
130
GeV
mHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
Obviously these are generalizations of MSSM.
We have more leeway for the DM, though.
• triplet
scalar DM
• singlet
• doublet
0.2
0.3
0.4
0.10.
106 108 1010 1012 1014 1016
115
120
125
130
GeV
mHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
0.2
0.3
0.4
0.10.
106 108 1010 1012 1014 1016
115
120
125
130
GeVmHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
0.2
0.3
0.4
0.10.
106 108 1010 1012 1014 1016
115
120
125
130
GeVmHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
• triplet
scalar DM
• singlet
• doublet
0.2
0.3
0.4
0.10.
106 108 1010 1012 1014 1016
115
120
125
130
GeV
mHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
0.2
0.3
0.4
0.10.
106 108 1010 1012 1014 1016
115
120
125
130
GeVmHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
0.2
0.3
0.4
0.10.
106 108 1010 1012 1014 1016
115
120
125
130
GeVmHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
assume Z2
assume PQ
RG equations
We coded up Machacek + Vaughn.
A.1 SM
b(1)g1 = 41g3110
b(2)g1 = −12g
31y
2b− 17
10g31y
2t− 3
2g31y
2τ +
199g5150 + 27
10g22g
31 +
445 g
23g
31
b(1)g2 = −19g326
b(2)g2 = −32g
32y
2b− 3
2g32y
2t− 1
2g32y
2τ +
35g526 + 9
10g21g
32 + 12g23g
32
b(1)g3 = −7g33
b(2)g3 = −2g33y2b− 2g33y
2t− 26g53 +
1110g
21g
33 +
92g
22g
33
b(1)yt = 32y
2byt − 17
20g21yt − 9
4g22yt − 8g23yt + yty2τ +
9y3t
2
b(2)yt = 780g
21y
2byt+
9916g
22y
2byt+4g23y
2byt+
54y
2byty2τ − 11
4 y2by3t− 1
4y4byt+
158 g
21yty
2τ +
158 g
22yty
2τ +
39380 g
21y
3t+
22516 g
22y
3t+ 36g23y
3t+ 1187
600 g41yt − 23
4 g42yt − 108g43yt − 9
20g21g
22yt +
1915g
21g
23yt + 9g22g
23yt − 6λHy3t +
32λ
2Hyt − 9
4y3ty2τ − 9
4yty4τ − 12y5
t
b(1)yb = −14g
21yb − 9
4g22yb − 8g23yb +
32yby
2t+ yby2τ +
9y3b
2
b(2)yb = 9180g
21yby
2t+ 99
16g22yby
2t+4g23yby
2t+ 15
8 g21yby
2τ +
158 g
22yby
2τ +
23780 g
21y
3b+ 225
16 g22y
3b+36g23y
3b− 127
600g41yb−
234 g
42yb − 108g43yb − 27
20g21g
22yb +
3115g
21g
23yb + 9g22g
23yb − 6y3
bλH + 3
2ybλ2H+ 5
4yby2ty2τ − 11
4 y3by2t−
14yby
4t− 9
4y3by2τ − 9
4yby4τ − 12y5
b
b(1)yτ = 3y2byτ − 9
4g21yτ − 9
4g22yτ + 3y2
tyτ +
5y3τ2
b(2)yτ = 58g
21y
2byτ+
458 g
22y
2byτ+20g23y
2byτ+
32y
2by2tyτ− 27
4 y2by3τ− 27
4 y4byτ+
178 g
21y
2tyτ+
458 g
22y
2tyτ+20g23y
2tyτ+
53780 g
21y
3τ +
16516 g
22y
3τ +
1371200 g
41yτ − 23
4 g42yτ +
2720g
21g
22yτ −6λHy3τ +
32λ
2Hyτ − 27
4 y2ty3τ − 27
4 y4tyτ −3y5τ
b(1)λH= 12y2
bλH−12y4
b− 9
5g21λH−9g22λH+ 27g41
100 + 910g
22g
21+
9g424 +12λHy2t +4λHy2τ +12λ2
H−12y4
t−4y4τ
b(2)λH= 5
2g21y
2bλH+ 45
2 g22y
2bλH+80g23y
2bλH+ 9
10g41y
2b+ 8
5g21y
4b+ 27
5 g22g
21y
2b−64g23y
4b− 9
2g42y
2b−42y2
bλHy2t −
72y2bλ2H−3y4
bλH−12y2
by4t−12y4
by2t+60y6
b+ 17
2 g21λHy2t +
452 g
22λHy2t +80g23λHy2t +
152 g
21λHy2τ +
152 g
22λHy2τ+
1887200 g
41λH+
545 g
21λ
2H+ 117
20 g22g
21λH+54g22λ
2H− 73
8 g42λH− 171
50 g41y
2t− 16
5 g21y
4t+ 63
5 g22g
21y
2t−
64g23y4t− 9
2g42y
2t− 9
2g41y
2τ − 24
5 g21y
4τ +
335 g
22g
21y
2τ − 3
2g42y
2τ −
3411g611000 − 1677
200 g22g
41 − 289
40 g42g
21 +
305g628 −
72λ2Hy2t− 3λHy4t − 24λ2
Hy2τ − λHy4τ − 78λ3
H+ 60y6
t+ 20y6τ
21
L 2-loop RGEs
do_physics.nb
boundary conditions
alone without including singlets or triplets. For the singlet and triplet theories we have includedall operators permitted by gauge symmetry subject to a discrete Z2 symmetry under which S
and T are odd. For the doublet scalar theory we have assumed a Peccei-Quinn symmetry whichacts oppositely on D and H in order to reduce the number of possible operators to a manageablenumber. Hence, all of these theories carry a Z2 symmetry under which the lightest odd particleis a prospective DM particle.
3 Methodology
To study vacuum stability and perturbativity in the presence of additional dynamics we haveproduced code which inputs an arbitrary Lagrangian and outputs the corresponding two-loopRGEs and one-loop weak scale threshold corrections to Higgs and top couplings. For the two-loopRGEs we employed the MS expressions of [11], including the corrections discussed in [12]. Wehave also computed one-loop weak scale threshold effects for the theories under considerationusing the methodology of [13]. All of these results, including the general formulae for thethreshold corrections using the notation of [12] and [14], are presented in detail in App. A andApp. B.
As is well-known, the running of λH is highly sensitive to the weak scale values of λH , yt andαs. This is because Higgs boson/top quark loops are the primary contribution driving λH to bepositive/negative in the ultraviolet. In turn, the strong interactions feed into the top Yukawacoupling. In our analysis we take the current values of the top quark pole mass and the MSrunning QCD coupling measured at the Z pole [15]:
mt = 173.2± 0.9 GeV (1σ) (7)
αs(mZ) = 0.1184± 0.0007 (1σ).
On the other hand the Higgs boson has not been found and its mass has yet to be measured.Nevertheless, we make use of recent experimental results from the LHC. Since almost all of thetheoretically allowed region for mH is excluded at 95% CL by both ATLAS and CMS, and bothcollaborations observe an excess at 125 - 126 GeV, assuming that the present excess correspondsto the Higgs boson, we will use
mH = 125 GeV, (8)
in our analysis. We relate the top quark and Higgs boson pole masses to yt(Q) and λH(Q), theYukawa and quartic couplings renormalized at an MS scale Q, via
mt = yt(Q)v[1 + δt(Q)]−1 (9)
m2H
= 2λ(Q)v2[1 + δH(Q)]. (10)
Here δt,H includes one-loop threshold corrections from the SM and new physics contributions.We present general formulae for the one-loop threshold corrections in App. B. Furthermore δt
8
alone without including singlets or triplets. For the singlet and triplet theories we have includedall operators permitted by gauge symmetry subject to a discrete Z2 symmetry under which S
and T are odd. For the doublet scalar theory we have assumed a Peccei-Quinn symmetry whichacts oppositely on D and H in order to reduce the number of possible operators to a manageablenumber. Hence, all of these theories carry a Z2 symmetry under which the lightest odd particleis a prospective DM particle.
3 Methodology
To study vacuum stability and perturbativity in the presence of additional dynamics we haveproduced code which inputs an arbitrary Lagrangian and outputs the corresponding two-loopRGEs and one-loop weak scale threshold corrections to Higgs and top couplings. For the two-loopRGEs we employed the MS expressions of [11], including the corrections discussed in [12]. Wehave also computed one-loop weak scale threshold effects for the theories under considerationusing the methodology of [13]. All of these results, including the general formulae for thethreshold corrections using the notation of [12] and [14], are presented in detail in App. A andApp. B.
As is well-known, the running of λH is highly sensitive to the weak scale values of λH , yt andαs. This is because Higgs boson/top quark loops are the primary contribution driving λH to bepositive/negative in the ultraviolet. In turn, the strong interactions feed into the top Yukawacoupling. In our analysis we take the current values of the top quark pole mass and the MSrunning QCD coupling measured at the Z pole [15]:
mt = 173.2± 0.9 GeV (1σ) (7)
αs(mZ) = 0.1184± 0.0007 (1σ).
On the other hand the Higgs boson has not been found and its mass has yet to be measured.Nevertheless, we make use of recent experimental results from the LHC. Since almost all of thetheoretically allowed region for mH is excluded at 95% CL by both ATLAS and CMS, and bothcollaborations observe an excess at 125 - 126 GeV, assuming that the present excess correspondsto the Higgs boson, we will use
mH = 125 GeV, (8)
in our analysis. We relate the top quark and Higgs boson pole masses to yt(Q) and λH(Q), theYukawa and quartic couplings renormalized at an MS scale Q, via
mt = yt(Q)v[1 + δt(Q)]−1 (9)
m2H
= 2λ(Q)v2[1 + δH(Q)]. (10)
Here δt,H includes one-loop threshold corrections from the SM and new physics contributions.We present general formulae for the one-loop threshold corrections in App. B. Furthermore δt
8
alone without including singlets or triplets. For the singlet and triplet theories we have includedall operators permitted by gauge symmetry subject to a discrete Z2 symmetry under which S
and T are odd. For the doublet scalar theory we have assumed a Peccei-Quinn symmetry whichacts oppositely on D and H in order to reduce the number of possible operators to a manageablenumber. Hence, all of these theories carry a Z2 symmetry under which the lightest odd particleis a prospective DM particle.
3 Methodology
To study vacuum stability and perturbativity in the presence of additional dynamics we haveproduced code which inputs an arbitrary Lagrangian and outputs the corresponding two-loopRGEs and one-loop weak scale threshold corrections to Higgs and top couplings. For the two-loopRGEs we employed the MS expressions of [11], including the corrections discussed in [12]. Wehave also computed one-loop weak scale threshold effects for the theories under considerationusing the methodology of [13]. All of these results, including the general formulae for thethreshold corrections using the notation of [12] and [14], are presented in detail in App. A andApp. B.
As is well-known, the running of λH is highly sensitive to the weak scale values of λH , yt andαs. This is because Higgs boson/top quark loops are the primary contribution driving λH to bepositive/negative in the ultraviolet. In turn, the strong interactions feed into the top Yukawacoupling. In our analysis we take the current values of the top quark pole mass and the MSrunning QCD coupling measured at the Z pole [15]:
mt = 173.2± 0.9 GeV (1σ) (7)
αs(mZ) = 0.1184± 0.0007 (1σ).
On the other hand the Higgs boson has not been found and its mass has yet to be measured.Nevertheless, we make use of recent experimental results from the LHC. Since almost all of thetheoretically allowed region for mH is excluded at 95% CL by both ATLAS and CMS, and bothcollaborations observe an excess at 125 - 126 GeV, assuming that the present excess correspondsto the Higgs boson, we will use
mH = 125 GeV, (8)
in our analysis. We relate the top quark and Higgs boson pole masses to yt(Q) and λH(Q), theYukawa and quartic couplings renormalized at an MS scale Q, via
mt = yt(Q)v[1 + δt(Q)]−1 (9)
m2H
= 2λ(Q)v2[1 + δH(Q)]. (10)
Here δt,H includes one-loop threshold corrections from the SM and new physics contributions.We present general formulae for the one-loop threshold corrections in App. B. Furthermore δt
8
The most important boundary conditions:
Computed 1-loop thresholds for the models:
stability and scalars
With add’l scalars, the vacuum structure is enriched. For example, singlet scalar DM:
0.2
0.3
0.4
0.10.
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115
120
125
130
GeV
mHGeV
1.62.1.2
0.8
106 108 1010 1012 1014 1016100
120
140
160
180
200
220
240
GeV
mHGeV
SM Doublet Scalar
Figure 5: Same as Fig. (3) but for SM + doublet scalar and varying κD with λD = λD = κ
D=0.
theories and the MSSM is
S ↔ B
T ↔ W T
D ↔ Hd
Dc ↔ Hu
mS ↔ M1
mT ↔ M2
mD ↔ µ
yS ↔ gd/√2
ycS ↔ g
u/√2
yT ↔ −gd/√2
ycT ↔ gu/
√2
, (5)
using the notation of [10]. Furthermore, in the limit of exact supersymmetry, g()u = g
()sin β
and g()d = g
()cos β.
For the case of new scalars we consider the following theories:
singlet scalar: −∆L =1
2mSS
2+
λS
2S4+
κS
2S2|H|2
triplet scalar: −∆L =1
2mTT
2+
λT
2T
4+
κT
2T
2|H|2
doublet scalar: −∆L = mD|D|2 + λD
2|D|4 + κD
2|D|2|H|2 + κ
D
2|DH
†|2.
(6)
Here S and T are real scalars while D is a complex scalar. Contrary to the fermionic case, the
pure doublet scalar case can have direct couplings to the Higgs and therefore can be considered
7
λH ≥ 0
λS ≥ 0
κS ≥ −2λHλS
absolutestability
non-perturbativity
We also consider when couplings blow up.
includes the shift between pole and running mass due to QCD interactions which is known tothree loops [17].
We have chosen our MS matching scale Q to be the top pole mass mt. Our boundaryconditions for SM parameters are determined by Eq. (8), Eq. (8), and Eq. (10), where we haverun αs(mZ) from mZ to mt using the SM two-loop RGEs enhanced by the three-loop QCD betafunction [15]. We then run the RGEs up to a scale Q = Λ and determine the value of λH atthat scale. As defined in Eq. (1), if λH < λH < 0 then the vacuum is metastable but long-lived,while if λH < λH then the vacuum is unstable and decays within the age of the Universe.
Note that λH ≥ 0 is a necessary but not sufficient condition for absolute stability of thevacuum. Likewise, 0 > λH > λH does not guarantee that the lifetime of the vacuum exceeds theage of the Universe. The reason for this is that in theories with new scalar fields, the vacuumstructure is enriched. Absolute stability of the vacuum implies further conditions on scalar fieldtheories: the self-quartic couplings satisfy λS,λT ,λD ≥ 0, while the cross-quartic couplings arebounded by
κS ≥ −2
λHλS (11)
κT ≥ −2
λHλT (12)
κD + κD
≥ −2
λHλD, (13)
for all scales below the cutoff Λ. If these criteria are not satisfied, then there will exist fielddirections in which the potential is unbounded from below at large field values. Alternatively, ifthe additional scalar fields acquire vacuum expectation values, this can also substantially alterthe stability of the vacuum [16]. However, we will not consider this possibility since our interestis in DM.
In addition to the question of stability we also investigate the perturbativity of interactionsin the ultraviolet. For our purposes we define perturbativity to be the criterion that for eachcoupling g, the contribution of g to its own beta function is bounded by unity. Precisely, werequire that dg/d logQ = β(g) < 1, which can be read off trivially from RGEs presented inApp. A. As we will see, the constraint of perturbativity will be largely unimportant for the caseof new fermionic states, but can play an essential role in determining the cutoff for theories withnew scalars. For the scalars, the perturbativity bounds on the scalar couplings are
κ2S< 16π2/4, λ2
S< 16π2/36 (14)
κ2T< 16π2/4, λ2
T< 16π2/44 (15)
κ2D,κ
D
2 < 16π2/2, λ2D< 16π2/12. (16)
To evaluate the properties of DM in each of these theories, we have implemented eachmodel in LanHEP [18] and evaluated relic abundances and direct detection cross-sections inmicrOMEGAs [19]. For the direct detection cross-sections we have employed the most recentlattice results [20] for the nuclear form factors,
f (p)Tu
= 0.0280 f (p)Td
= 0.0280 f (p)Ts
= 0.0689. (17)
9
where the threshold is defined from when the RHS of the RGEs diverge.
results
Higgs-fermion couplings tend to destabilize.
0.40.50.6
0.3
0.2
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140
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SM SingletDoublet Fermion
0.50.6
0.3
0.2
0.4
104 106 108 1010 1012115
120
125
130
135
140
GeVmHGeV
SM TripletDoublet Fermion
Figure 2: Metastability bands for SM + singlet/doublet and triplet/doublet fermion, shown asa function of the Higgs mass. Regions above/below each band are stable/unstable. Each bandis labeled with the corresponding value of the Yukawa coupling, yS,T = yc
S,T. The dashed lines
correspond to the value of mH suggested by [5, 6].
summarize the nature of the vacuum by depicting the “metastability band” in parameter spacedefined by the region
λH < λH < 0. (2)
Note that for theories in which supersymmetric dynamics enters at the cutoff Λ, i.e. split orhigh-scale supersymmetry, absolute stability is required by the fact that the potential is positivesemi-definite in all field directions, so λH ≥ 0.
By running the RGEs into the ultraviolet, subject to infrared boundary conditions, it ispossible to determine the scale Λ at which the quartic coupling crosses through the metasta-bility band. For example, in Fig. (1) we plot the scale Λ indicating the onset of stabil-ity/metastability/instabilility in the SM as a function of the mH . Our results are in niceagreement with existing calculations in the literature [2, 3, 4].
The remainder of this paper is as follows. In Sec. 2 we briefly summarize the set of modelsto be studied and establish notational conventions. We present our methodology in Sec. 3regarding the running of couplings and evaluation of DM properties. Finally, in Sec. 4 wediscuss our results. The two-loop RGEs and one-loop threshold corrections for these theoriesare presented in App. A and App. B.
4
Higgs-fermion couplings tend to destabilize.
0.40.50.6
0.3
0.2
106 108 1010 1012 1014115
120
125
130
135
140
GeV
mHGeV
SM SingletDoublet Fermion
0.50.6
0.3
0.2
0.4
104 106 108 1010 1012115
120
125
130
135
140
GeVmHGeV
SM TripletDoublet Fermion
Figure 2: Metastability bands for SM + singlet/doublet and triplet/doublet fermion, shown asa function of the Higgs mass. Regions above/below each band are stable/unstable. Each bandis labeled with the corresponding value of the Yukawa coupling, yS,T = yc
S,T. The dashed lines
correspond to the value of mH suggested by [5, 6].
summarize the nature of the vacuum by depicting the “metastability band” in parameter spacedefined by the region
λH < λH < 0. (2)
Note that for theories in which supersymmetric dynamics enters at the cutoff Λ, i.e. split orhigh-scale supersymmetry, absolute stability is required by the fact that the potential is positivesemi-definite in all field directions, so λH ≥ 0.
By running the RGEs into the ultraviolet, subject to infrared boundary conditions, it ispossible to determine the scale Λ at which the quartic coupling crosses through the metasta-bility band. For example, in Fig. (1) we plot the scale Λ indicating the onset of stabil-ity/metastability/instabilility in the SM as a function of the mH . Our results are in niceagreement with existing calculations in the literature [2, 3, 4].
The remainder of this paper is as follows. In Sec. 2 we briefly summarize the set of modelsto be studied and establish notational conventions. We present our methodology in Sec. 3regarding the running of couplings and evaluation of DM properties. Finally, in Sec. 4 wediscuss our results. The two-loop RGEs and one-loop threshold corrections for these theoriesare presented in App. A and App. B.
4
Higgs-scalar couplings tend to stabilize.
0.2
0.3
0.4
0.10.
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GeVmHGeV
SM Singlet Scalar
Figure 3: Higgs mass bounds as a function of the cutoff, for the SM + singlet scalar. The left
panel depicts metastability bands and the right panel depicts the scale at which the largest
coupling in the theory becomes non-perturbative. Each label denotes the corresponding value
of the cross-quartic coupling, κS, and we have fixed the self-quartic coupling, λS = 0.
2 Models
In this section we briefly summarize the models we will analyze and provide notational conven-
tions. We choose the following normalization for the SM quartic and Yukawa couplings,
− L = ytqHtc+ ybqH
†bc+ yτH
†τ c +λH
2(|H|2 − v
2)2, (3)
where v = 174.1 GeV is the Higgs vacuum expectation value. Throughout, we ignore the effectsof the light fermion generations because they are negligible for our results.
We will augment the SM with new states, sending L → L +∆L. Throughout, if a particle
X is a fermion, then yX will denote its Yukawa coupling of X to the Higgs. If a particle X
is a scalar, then λX will denote its self-quartic coupling, while κX will denote its cross-quartic
coupling to the Higgs. Finally, the mass parameter for the particle X will be denoted by mX .
5
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 250 GeV, mD 500 GeV
104
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1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
yScy S
mS 250 GeV, mD 750 GeV
104
1010
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1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 750 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 1000 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 6: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(ycS, yS) plane at fixed values of mS and mD. The purple bands corresponds to Ωh2 = 0.11±0.01.The red/blue regions are excluded/allowed by XENON100, with σSI denoted. The gray contours
in the upper left/lower right quadrants denote the scale Λ (GeV) at which the vacuum becomes
metastable/unstable.
11
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y SmS 250 GeV, mD 500 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 250 GeV, mD 750 GeV
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 750 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y SmS 500 GeV, mD 1000 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 6: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(ycS, yS) plane at fixed values of mS and mD. The purple bands corresponds to Ωh2 = 0.11±0.01.The red/blue regions are excluded/allowed by XENON100, with σSI denoted. The gray contours
in the upper left/lower right quadrants denote the scale Λ (GeV) at which the vacuum becomes
metastable/unstable.
11
singlet/doubletfermion DM
directdetection
thermalrelic
window for the cutoff + =
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 250 GeV, mD 500 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y SmS 250 GeV, mD 750 GeV
104
1010
104
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1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 750 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 1000 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 6: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(ycS, yS) plane at fixed values of mS and mD. The purple bands corresponds to Ωh2 = 0.11±0.01.The red/blue regions are excluded/allowed by XENON100, with σSI denoted. The gray contours
in the upper left/lower right quadrants denote the scale Λ (GeV) at which the vacuum becomes
metastable/unstable.
11
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 250 GeV, mD 500 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
yScy S
mS 250 GeV, mD 750 GeV
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 750 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 1000 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 6: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(ycS, yS) plane at fixed values of mS and mD. The purple bands corresponds to Ωh2 = 0.11±0.01.The red/blue regions are excluded/allowed by XENON100, with σSI denoted. The gray contours
in the upper left/lower right quadrants denote the scale Λ (GeV) at which the vacuum becomes
metastable/unstable.
11
directdetection
thermalrelic
window for the cutoff + =
increase doublet mass
singlet/doubletfermion DM
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 250 GeV, mD 500 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 250 GeV, mD 750 GeV
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y SmS 500 GeV, mD 750 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 1000 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 6: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(ycS, yS) plane at fixed values of mS and mD. The purple bands corresponds to Ωh2 = 0.11±0.01.The red/blue regions are excluded/allowed by XENON100, with σSI denoted. The gray contours
in the upper left/lower right quadrants denote the scale Λ (GeV) at which the vacuum becomes
metastable/unstable.
11
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 250 GeV, mD 500 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
yScy S
mS 250 GeV, mD 750 GeV
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 750 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 1000 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 6: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(ycS, yS) plane at fixed values of mS and mD. The purple bands corresponds to Ωh2 = 0.11±0.01.The red/blue regions are excluded/allowed by XENON100, with σSI denoted. The gray contours
in the upper left/lower right quadrants denote the scale Λ (GeV) at which the vacuum becomes
metastable/unstable.
11
directdetection
thermalrelic
window for the cutoff + =
increase both masses
singlet/doubletfermion DM
104
1010
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1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
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0.0
0.5
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ySc
y S
mS 250 GeV, mD 500 GeV
104
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1010
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1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
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1.0
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ySc
y S
mS 250 GeV, mD 750 GeV
104
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1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y SmS 500 GeV, mD 750 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 1000 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 6: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(ycS, yS) plane at fixed values of mS and mD. The purple bands corresponds to Ωh2 = 0.11±0.01.The red/blue regions are excluded/allowed by XENON100, with σSI denoted. The gray contours
in the upper left/lower right quadrants denote the scale Λ (GeV) at which the vacuum becomes
metastable/unstable.
11
104
1010
104
1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 250 GeV, mD 500 GeV
104
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1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
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yScy S
mS 250 GeV, mD 750 GeV
104
1010
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1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
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0.0
0.5
1.0
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ySc
y S
mS 500 GeV, mD 750 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 1000 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 6: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(ycS, yS) plane at fixed values of mS and mD. The purple bands corresponds to Ωh2 = 0.11±0.01.The red/blue regions are excluded/allowed by XENON100, with σSI denoted. The gray contours
in the upper left/lower right quadrants denote the scale Λ (GeV) at which the vacuum becomes
metastable/unstable.
11
directdetection
thermalrelic
window for the cutoff + =
increase both masses
could be relevant to indirect searches
singlet/doubletfermion DM
104
1010
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1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
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1.0
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ySc
y S
mS 1375 GeV, mD 1125 GeV
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y S
mS 1625 GeV, mD 1125 GeV
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1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
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ySc
y S
mS 1500 GeV, mD 1250 GeV
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1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 1750 GeV, mD 1250 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 7: Same as Fig. (6) but for SM + singlet/doublet fermion with doublet-like DM.
12
... likewise for fermion doublet-like DM, etc...
in terms of observables...
Singlet/doublet fermion DM has 4 params.:
mS
mD
yS
ycS
σSI
Ωh2
m0
ycS/yS
Remap results into “physical” param. space.
104
1010
200 400 600 800 10000
200
400
600
800
1000
m0 GeVmm0GeV
tan1yScyS 5°
104
1010
200 400 600 800 10000
200
400
600
800
1000
m0 GeV
mm0GeV
tan1yScyS 10°
104
1010
200 400 600 800 10000
200
400
600
800
1000
m0 GeV
mm0GeV
tan1yScyS 15°
104
1010
200 400 600 800 10000
200
400
600
800
1000
m0 GeVmm0GeV
tan1yScyS 20°
1046 1045 1044 1043
ΣSI cm2
Figure 10: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(m0,m+ −m0) plane at fixed values of ycS/yS. All points are consistent with Ωh2 = 0.11, whilethe red/blue regions are excluded/allowed by XENON100, with σSI denoted. The solid gray
contours denote the scale Λ (GeV) at which the vacuum becomes metastable.
16
104
1010
104
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1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 250 GeV, mD 500 GeV
104
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1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
yScy S
mS 250 GeV, mD 750 GeV
104
1010
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1016
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 750 GeV
104
1016
1010
104
1.5 1.0 0.5 0.0 0.5 1.0 1.51.5
1.0
0.5
0.0
0.5
1.0
1.5
ySc
y S
mS 500 GeV, mD 1000 GeV
1046 1045 1044 1043
ΣSI cm2
Figure 6: Contour plots for SM + singlet/doublet fermion with singlet-like DM, shown in the
(ycS, yS) plane at fixed values of mS and mD. The purple bands corresponds to Ωh2 = 0.11±0.01.The red/blue regions are excluded/allowed by XENON100, with σSI denoted. The gray contours
in the upper left/lower right quadrants denote the scale Λ (GeV) at which the vacuum becomes
metastable/unstable.
11
104
106
108
1010
1012
1014
0.3 0.2 0.1 0.0 0.1 0.2 0.30.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
ΚS
Λ S
0.3 0.2 0.1 0.0 0.1 0.2 0.3
200
400
600
800
1045.4
1045.3
1045.2
ΚS
mSGeV
ΣSIcm2
SM Singlet Scalar
Figure 11: Plots for SM + singlet scalar DM, shown as a function of the quartic couplings,
κS and λS. The left panel depicts contours Λ (GeV) required by stability and perturbativity.
The red/blue/purple regions correspond to regimes in which Λ is determined by metastability
due to a negative Higgs quartic/metastability due to a large and negative cross-quartic/non-
perturbative couplings. The right panel depicts the value of mS required for a thermal relic
abundance along with the predicted σSI .
104106
108
1010
1012
1014
1016
0.4 0.3 0.2 0.1 0.0 0.1 0.20.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
ΚT
Λ T
0.4 0.3 0.2 0.1 0.0 0.1 0.21860
1880
1900
1920
1940
1960
1980
1050
1049
1048
1047
1046
ΚT
mTGeV
ΣSIcm2
SM Triplet Scalar
Figure 12: Same as Fig. (11) but for SM + triplet scalar DM.
17
For scalars, perturbativity is important.
104
106
108
1010
1012
1014
0.3 0.2 0.1 0.0 0.1 0.2 0.30.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
ΚS
Λ S
0.3 0.2 0.1 0.0 0.1 0.2 0.3
200
400
600
800
1045.4
1045.3
1045.2
ΚS
mSGeV
ΣSIcm2
SM Singlet Scalar
Figure 11: Plots for SM + singlet scalar DM, shown as a function of the quartic couplings,
κS and λS. The left panel depicts contours Λ (GeV) required by stability and perturbativity.
The red/blue/purple regions correspond to regimes in which Λ is determined by metastability
due to a negative Higgs quartic/metastability due to a large and negative cross-quartic/non-
perturbative couplings. The right panel depicts the value of mS required for a thermal relic
abundance along with the predicted σSI .
104106
108
1010
1012
1014
1016
0.4 0.3 0.2 0.1 0.0 0.1 0.20.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
ΚT
Λ T
0.4 0.3 0.2 0.1 0.0 0.1 0.21860
1880
1900
1920
1940
1960
1980
1050
1049
1048
1047
1046
ΚT
mTGeV
ΣSIcm2
SM Triplet Scalar
Figure 12: Same as Fig. (11) but for SM + triplet scalar DM.
17
For scalars, perturbativity is important.
κT ≥ −2λHλT λH ≥ 0
non-perturbative
• Conservative implications of 125 GeV should be derived for the SM plus DM!
• Scalar DM stabilizes the vacuum, but non-perturbativity can instead impose a cutoff.
• Fermion DM destabilizes the vacuum, so an add’l “oasis” is often needed.
conclusions