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 minimum 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, although 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 minimum 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, although 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


• strong CP


• neutrino masses

• dark matter

tuningdynamics


• flavor puzzle


• strong CP


• neutrino masses

• dark matter

tuningdynamics


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.

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




2mSS

2+mDDD

c+ ySHSD + y

cSH

cSD

c


2mTT

2+mDDD

c+ yTHTD + y

cTH

cTD

c,

(4)












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




2mSS

2+mDDD

c+ ySHSD + y

cSH

cSD

c


2mTT

2+mDDD

c+ yTHTD + y

cTH

cTD

c,

(4)












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




2mSS

2+mDDD

c+ ySHSD + y

cSH

cSD

c


2mTT

2+mDDD

c+ yTHTD + y

cTH

cTD

c,

(4)












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


D=0.


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)


()sin β

and g()d = g

()cos β.



2mSS

2+

λS

2S4+

κS

2S2|H|2


2mTT

2+

λT

2T

4+

κT

2T

2|H|2


2|D|4 + κD

2|D|2|H|2 + κ

D

2|DH

†|2.

(6)



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


D=0.


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)


()sin β

and g()d = g

()cos β.



2mSS

2+

λS

2S4+

κS

2S2|H|2


2mTT

2+

λT

2T

4+

κT

2T

2|H|2


2|D|4 + κD

2|D|2|H|2 + κ

D

2|DH

†|2.

(6)



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


D=0.


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)


()sin β

and g()d = g

()cos β.



2mSS

2+

λS

2S4+

κS

2S2|H|2


2mTT

2+

λT

2T

4+

κT

2T

2|H|2


2|D|4 + κD

2|D|2|H|2 + κ

D

2|DH

†|2.

(6)



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


D=0.


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)


()sin β

and g()d = g

()cos β.



2mSS

2+

λS

2S4+

κS

2S2|H|2


2mTT

2+

λT

2T

4+

κT

2T

2|H|2


2|D|4 + κD

2|D|2|H|2 + κ

D

2|DH

†|2.

(6)



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


D=0.


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)


()sin β

and g()d = g

()cos β.



2mSS

2+

λS

2S4+

κS

2S2|H|2


2mTT

2+

λT

2T

4+

κT

2T

2|H|2


2|D|4 + κD

2|D|2|H|2 + κ

D

2|DH

†|2.

(6)



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


D=0.


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)


()sin β

and g()d = g

()cos β.



2mSS

2+

λS

2S4+

κS

2S2|H|2


2mTT

2+

λT

2T

4+

κT

2T

2|H|2


2|D|4 + κD

2|D|2|H|2 + κ

D

2|DH

†|2.

(6)



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


D=0.


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)


()sin β

and g()d = g

()cos β.



2mSS

2+

λS

2S4+

κS

2S2|H|2


2mTT

2+

λT

2T

4+

κT

2T

2|H|2


2|D|4 + κD

2|D|2|H|2 + κ

D

2|DH

†|2.

(6)



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


D=0.


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)


()sin β

and g()d = g

()cos β.



2mSS

2+

λS

2S4+

κS

2S2|H|2


2mTT

2+

λT

2T

4+

κT

2T

2|H|2


2|D|4 + κD

2|D|2|H|2 + κ

D

2|DH

†|2.

(6)



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



3 Methodology



mt = 173.2± 0.9 GeV (1σ) (7)

αs(mZ) = 0.1184± 0.0007 (1σ).


mH = 125 GeV, (8)


mt = yt(Q)v[1 + δt(Q)]−1 (9)

m2H

= 2λ(Q)v2[1 + δH(Q)]. (10)


8



3 Methodology



mt = 173.2± 0.9 GeV (1σ) (7)

αs(mZ) = 0.1184± 0.0007 (1σ).


mH = 125 GeV, (8)


mt = yt(Q)v[1 + δt(Q)]−1 (9)

m2H

= 2λ(Q)v2[1 + δH(Q)]. (10)


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.

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


D=0.


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)


()sin β

and g()d = g

()cos β.



2mSS

2+

λS

2S4+

κS

2S2|H|2


2mTT

2+

λT

2T

4+

κT

2T

2|H|2


2|D|4 + κD

2|D|2|H|2 + κ

D

2|DH

†|2.

(6)



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

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.

106 108 1010 1012 1014 1016 1018

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

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

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


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


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


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


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


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


1046 1045 1044 1043

ΣSI cm2





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


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


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


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


1046 1045 1044 1043

ΣSI cm2





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


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


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


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


1046 1045 1044 1043

ΣSI cm2





11

directdetection

thermalrelic


increase doublet mass


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


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


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


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


1046 1045 1044 1043

ΣSI cm2





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


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


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


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


1046 1045 1044 1043

ΣSI cm2





11

directdetection

thermalrelic


increase both masses


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


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


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


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


1046 1045 1044 1043

ΣSI cm2





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


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


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


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


1046 1045 1044 1043

ΣSI cm2





11

directdetection

thermalrelic


increase both masses

could be relevant to indirect searches


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


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


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


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


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


(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

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


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


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


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


1046 1045 1044 1043

ΣSI cm2





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

higgs and dark matter hints of an oasis in the...

Documents