eichhorn holger

Higgs-Dark Matter Connection
and the Scale of New Physics
Michael M. Scherer
Institute for Theoretical Physics, University of Heidelberg
in collaboration with
Astrid Eichhorn, Holger Gies, Joerg Jaeckel, Tilman Plehn and René Sondenheimer
June 10, 2015 @ WIN2015, Heidelberg
JHEP 04 (2015) 022 or arXiv:1501.02812
Phys. Rev. D 90 (2014) 025023 or arXiv:1404.5962
[Brout, Englert; Higgs; Guralnik, Hagen, Kibble, 1964]
The Standard Model and the Higgs
r0#~#1/me#
•
Discovery of the [Brout,
HiggsEnglert;
@ LHC:
Higgs; Guralnik, Hagen, Kibble, 1964]
ATLAS collaboration (2012)
[ATLAS collaboration, 2012]
[ATLAS collaboration,
2012] MH ≈
CMS collaboration (2012)
[CMS collaboration, 2012]
[CMS collaboration, 2012]
125 GeV
[Brout, Englert; Higgs; Guralnik, Hagen, Kibble, 1964]
The Standard Model and the Higgs
r0#~#1/me#
•
Discovery of the [Brout,
HiggsEnglert;
@ LHC:
Higgs; Guralnik, Hagen, Kibble, 1964]
ATLAS collaboration (2012)
[ATLAS collaboration, 2012]
[ATLAS collaboration,
2012] MH ≈
•
Standard model:
‣ effective theory
‣ physical cutoff ⇤
‣ “New Physics” beyond ⇤
CMS collaboration (2012)
[CMS collaboration, 2012]
[CMS collaboration, 2012]
125 GeV
[Brout, Englert; Higgs; Guralnik, Hagen, Kibble, 1964]
The Standard Model and the Higgs
r0#~#1/me#
•
Discovery of the [Brout,
HiggsEnglert;
@ LHC:
Higgs; Guralnik, Hagen, Kibble, 1964]
ATLAS collaboration (2012)
CMS collaboration (2012)
[ATLAS collaboration, 2012]
[ATLAS collaboration,
2012] MH ≈
•
Standard model:
‣ effective theory
‣ physical cutoff ⇤
‣ “New Physics” beyond ⇤
•
[CMS collaboration, 2012]
[CMS collaboration, 2012]
125 GeV
Range of validity of SM?
p
= ~c/G ⇡ 1019 GeV
‣ Gravity effects: ⇤ ⇠ MPl
‣ Landau pole in U(1)hypercharge:
‣ Higgs potential…
⇤ > MPl
Higgs Mass Bounds
Hambye & Riesselmann (1997)
Krive, Linde (1976)
Maiani, Parisi, Petronzio (1978)
Krasnikov (1978)
Politzer, Wolfram (1978)
Hung (1979)
Lindner (1985)
mh
Wetterich (1987)
Lindner, Sher, Zaglauer (1989)
Ford, Jones, Stephenson, Einhorn (1993)
Altarelli, Isidori (1994)
Schrempp, Wimmer (1996)…
125
ure 2: Summary of the uncertainties connected to the bounds on MH . p
The upper
2 4 ·for
vev
mass
related
to Higgs uncertainties
coupling andinvev:
• Higgs
h = bound
d area
indicates
theissum
of theoretical
the M mupper
H
= 175Upper
[12]. The
upper
corresponds
related
to edge
Landau
pole to Higgs masses for which the
• GeV bound
Higgs sector ceases to be meaningful at scale Λ (see text), and the lower edge
Mechanism for Lower Higgs Mass Bound
• Higgs potential:
µ 2
4
H + H4
2
4
U
U
SSB
vev
p
2 4 · vev
y
mt = p · vev
2
mh =
Mechanism for Lower Higgs Mass Bound
• Higgs potential:
µ 2
4
H + H4
2
4
U
2 4 · vev
y
mt = p · vev
2
mh =
U
SSB
p
vev
• Running Higgs self-coupling:
4
=-
+
top loop
+
Higgs loop
gauge contributions
Mechanism for Lower Higgs Mass Bound
• Higgs potential:
µ 2
4
H + H4
2
4
U
2 4 · vev
y
mt = p · vev
2
mh =
U
SSB
p
vev
• Running Higgs self-coupling:
4
=-
+
top loop
+
gauge contributions
Higgs loop
Higgs loop
4
• Choose
= 0 at ⇤UV :
➡ minimal value of Higgs mass
top loop
➡ for smaller mh: start with unstable potential?!
kEW
RG flow
⇤UV RG scale
4
a3 HMZ L = 0.1184 ± 0.0007HredL
MhRG
= 125.1
± 0.2
GeV
HblueL
scale
k in
GeV
Beta function of the Higgs quartic bl
SM couplings
running couplings
The beta function for any coupling g is defined as g = dg/d log k. In these conventions the
lization group equations
the Higgs
self-coupling
the top YukawaModel
y, and the strong coupl
LowerforMass
Bound
in the
4 , Standard
dard Model read

1
3
3
d 4
2
2
4
2
2
4
2
2 2
=
=
12
+
6
y
3y
3g
+
g
+
2g
+
(g
+
g
4
4
4
2
1
2
2
1)
4
2
d log k
8⇡
2
16

dy
y
9 2
9 2 17 2
2
=
y
8gs
g2
g1
y =
2
instability
d log k
16⇡ 2
4
12• Vacuum
1.0
1.0
 Buttazzo et al. (2013)
y
µ 2
3
4
4
d
g
g
2
H
+
H
g
crosses
zero
in
3s
s
yt
‣
4
2
4
0.8
=
11
n
.
0.8
gs =
f
2
d log k
16⇡
3
instability of Higgs vacuum
➡
0.6
g
2
0.6
p
2
10
Yukawa coupling
is
linked
to
the
top
mass
as
y
=
2m
/v
while
the
Higgs
mass
is
given
by
m
‘Scale
of
New
Physics’
~
10
GeV
t
‣ 0.4
H
g1
0.4
tributions
from dimension-6 operators. [Typically, these coupling parameters or the related
mt and mH , are evaluated at the scale of the top pole
be translated
into the pole m
0.2mass to
strongly
depends
on
top
Yukawa:
‣
0.2
gs and the
top, according to mthe
on-shell scheme.] The two lgauge
couplings are g2 for SU (2)L a
in TeV
0.10
4
0
l
Buttazzo
et al.g(2013)
0.0
3
(1)Y . The
number
of
fermions
contributing
to
the
running
of
the
strong
coupling
is
n
.
In
the
f
0.0 yb
3s bands in
0.08
± 0.8
GeV HgrayL14 by Standar
o explicit10higher
dimensional operators are present. However,
if105they 10Mare
generated
8 = 173.3
2
100
1011
10
1017
10
104 106 108 1010 1012 1014 1016 1018 1020
RGE scale m in GeV
Higgs quartic coupling l
t
0.06
0.04
(MPl ) =0.20.6168
running couplings
running couplings
1.0
enormalisation
of the SM gauge couplings g1 = 5/3gY , g2 , g3 , of the top,1.0bottom
0.02
Lowerg3Higgs
mass bound
ngs (yt , y•b , y ), ofy the
Higgs quartic coupling and of the Higgs mass parameter m. g3
0.8
0.8
y
ers are defined in the ms scheme. We include two-loop thresholds at the weak scale
0.00
op RG equations. m
The (M
thickness
indicates the ±1 uncertainties in Mt , Mh , 3 .
0.6
0.6
h
Pl ) ⇡ 129GeV
-0.02
exp
m
125GeV
s, we find 0.4
the following values
SM parameters:
0.4 -0.04
h of⇡the
2
10
(56a)
0.2
10
4
Mt = 171.1 GeV
as HMZ L = 0.1205
as HMZ L = 0.1163
-0
-0
-0
Mt = 175.6 GeV
-0
10
6
8
10
10
10
10
12
14
10
RGE scale m in GeV
10
16
10
18
10
20
Scenarios at the Scale of New Physics
@ ~ 1010 GeV several scenarios are possible:
1. New degrees of freedom appear that render Higgs potential stable - dark matter?
r are
runalidysics
perticle
data,
phe-
both the scale k as well as the field value, such that
@ ~ 1010 GeV several scenarios are possible:
1. New degrees of freedom appear that
V (k = ⇤; H) = VUV
and
V (k = 0; H) =
In this manner, the (meta-)stability properties of the e↵ective pot
render
Higgs potential stable - dark matter?
in a scale-dependent manner.
Gabrielli et al. (2013)
1040
2. Stable minimum might appear for large fieldFinally,
valueslet
➡
True minimum @ large H (<
us note another subtlety in the comparison betw
and non-perturbative-10
approaches. The perturbative approach usu
-10
-10
-10
scheme. To avoid large threshold e↵ect
MPl)?independent regularization
-10
stop the running of the
couplings at the appropriate mass scales.
-10
masses are proportional
to H. This suggests to approximate
40
50
60
70
VHhLêGeV4
➡
Metastability of Higgs vacuum?
80
90
-10100
➡
Small tunnelling rates to stable minimum?
Ve↵ = V (k = H, H) ⇡
5
10
15
1
4
20
4 (H) H
4
,
25
as2is usually done in the perturbative approach.
In Eq.(2.12) we h
log HhêGeVL
the running Higgs self-coupling at the scale kEW . Using the ide
FIG. 2: The SM Higgs e↵ective potential as a function of the
Gabrielli et al. (2013) essentially reproduce
2 2
termV (h)
in the
ofHiggs
the sec
Higgs the
fieldfirst
strength,
= µsquare
h + Hbrackets
(h) h4 . The
parameter is approximated as a constant and the runLet us check thismass
explicitly,
ning quartic coupling is evaluated at 1-loop level, where the
10
1.2
1.0
g3
SM Couplings
We
d to
new
ctor.
atument
Scenarios at the Scale of New Physics
Ve↵ (kEW ) ⇡
0.8
yt
0.6
g2
=
g1
✓
0.4
0.2
0.0
l
5
scale is set by the field strength h. The global minimum at
1026 GeV
VUV⇠ +
Vtop is generated by H running positive (from low
to high energy) at this scale. The local minimum
at the elec✓
◆
h
2
2
µ2 (⇤)
c
⇤
1
2⇤
troweak scale
by the negative
2 is caused
4 4 Higgs mass parameter.
H2 +
y
H log 1 + 2 2 +
2
64⇡ 2
y H
10
15
20
log10 HmêGeVL
25
30
35
40
FIG. 1: Running of the gauge couplings, the top Yukawa and
the Higgs self-coupling in the standard model. The Higgs
y2H 2
◆i
2
4 (⇤) 4
4
4⇤
log
1
+
+
H scale). Howtoward negative values2 (from high to low
2⇤
4
ever, the SM contains also the top quark with its large
✓
◆
hthe Higgs boson.
2
Yukawa
coupling
to
As
explained
in
µ2 (k
)
1
2⇤
EW
2
4 4
=
+ at 2lowy energies
H log near
1+
+ the
2y 2 ⇤
the lastH
section,
the 2EW2 scale
2 Yukawa becomes
64⇡ large and starts dominating
y H the H
top
✓
◆
running, pushing it
values. In the SM
y 2back
H 2 toipositive
4 (⇤) 4
4
the
at low
4⇤second
log local
1 +minimum
+ energies
His obtained by
2
2⇤
4 term µ2 to the
adding an explicit negative
Higgs mass

✓ ◆
Scenarios at the Scale of New Physics
both the scale k as well as the field value, such that
@ ~ 1010 GeV several scenarios are possible:
1. New degrees of freedom appear that
and
V (k = 0; H) =
In this manner, the (meta-)stability properties of the e↵ective pot
render
Higgs potential stable - dark matter?
in a scale-dependent manner.
Gabrielli et al. (2013)
1040
2. Stable minimum might appear for large fieldFinally,
valueslet
➡
True minimum @ large H (<
us note another subtlety in the comparison betw
and non-perturbative-10
approaches. The perturbative approach usu
-10
-10
-10
scheme. To avoid large threshold e↵ect
MPl)?independent regularization
-10
stop the running of the
couplings at the appropriate mass scales.
-10
masses are proportional
to H. This suggests to approximate
40
50
60
70
VHhLêGeV4
➡
Metastability of Higgs vacuum?
80
90
-10100
➡
Small tunnelling rates to stable minimum?
Ve↵ = V (k = H, H) ⇡
5
10
15
1
4
20
4
4 (H) H
,
25
as2is usually done in the perturbative approach.
In Eq.(2.12) we h
log HhêGeVL
the running Higgs self-coupling at the scale kEW . Using the ide
FIG. 2: The SM Higgs e↵ective potential as a function of the
6
8
Gabrielli
et
al.
(2013)
2 2
Higgs field (e.g.essentially
~H , H reproduce
, …) Higgs
to render
potential
stable
the
term
in the
ofHiggs
the sec
fieldfirst
strength,
V (h)
= µsquare
h + Hbrackets
(h) h4 . The
parameter is approximated as a constant and the runLet us check thismass
explicitly,
ning quartic coupling is evaluated at 1-loop level, where the
10
1.2
Include
higher powers in
1.0
➡
SM Couplings
We 3.
d to
new
ctor.
atument
r are
runalidysics
perticle
data,
phe-
V (k = ⇤; H) = VUV
0.8
is set by
the field strength h.
g
Do
not appear in perturbatively renormalizablescale
Higgs
Lagrangian
3
Ve↵ (kEW ) ⇡
yt
➡
0.6
0.4
➡
0.2
➡
0.0
The global minimum at
10 GeV
VUV⇠ +
Vtop is generated by H running positive (from low
to high energy) at this scale. The local minimum
at the elec✓
◆
h
2
2
µ2 (⇤)
c
⇤
1
2⇤
troweak scale
by the negative
2 is caused
4 4 Higgs mass parameter.
H2 +
y
H log 1 + 2 2 +
2
64⇡ 2
y H
26
g
Appear
in effective theories with finite ΛUV when
approaching underlying theory
=
2
g1
✓
y2H 2
◆i
10 GeV
4 (⇤) 4
New physics appears at higher scales 10? GeVtoward
> 410
4⇤
log
1
+
+
H scale).
negative values2 (from high to low
l
Link
to
particle
models?
5
10 BSM
15
20
25 physics
30
35
40
log10 HmêGeVL
FIG. 1: Running of the gauge couplings, the top Yukawa and
the Higgs self-coupling in the standard model. The Higgs
How-
2
2⇤ also the top 4quark with its large
ever, the SM contains
✓
◆
hthe Higgs boson.
2
Yukawa
coupling
to
As
explained
in
µ2 (k
)
1
2⇤
EW
2
4 4
=
+ at 2lowy energies
H log near
1+
+ the
2y 2 ⇤
the lastH
section,
the 2EW2 scale
2 Yukawa becomes
64⇡ large and starts dominating
y H the H
top
✓
◆
running, pushing it
values. In the SM
y 2back
H 2 toipositive
4 (⇤) 4
4
the
at low
4⇤second
log local
1 +minimum
+ energies
His obtained by
2
2⇤
4 term µ2 to the
adding an explicit negative
Higgs mass

✓ ◆
Higgs Portal to Dark Matter
Only gravitational interaction?
•
Cline et al. (2013)
Evidence for DM: gravitational lensing, galaxy rotation
curves, CMB, …
Higgs portal to dark matter
⇣
R 4 1(WIMP)µ
field serves as stable DM candidate
• Single scalarscalar,
gauge singlet: = d x 2 @µ @ +
DM
‣ with
Z2
Z
⇣1
⌘
1
02 4
= d4 x @µ S@ µ S + m2S S 2 +
S
Z2 -symmetry
! 2 : stable8 WIMP
2
m2
2
2
+
04
8
4
⌘
[Silveira, Zee, 1985; McDonald, 1994; Burgess, Pospelov, Veldhuis, 2001...]
symmetry:
S!
S
to Higgs: 11interaction?
h2 S 2
Only gravitational
! Coupling to Higgs possible:
‣ Portal coupling
4
χ!part!of!
• S can reproduce observed dark matter relic density
‣ Condition on scattering cross section
‣ Relation between mS and 11
‣ For mS > mh /2 :
log10
11
2
=
3.63 + 1.04 log10
dm!relic!!
density!
χ!over7
produced!
mS
GeV
11
4
2
Cline et al. (2013)
[Cline, Scott, Kainulainen, Weniger, 2013]
Effect of Dark Matter on Higgs Mass
top
• Running Higgs self-coupling:
Higgs
+
=-
DM fluctuations
+
without DM
with DM
11
⇤EW
RG flow
⇤UV
> 0: lower Higgs mass
Effect of Dark Matter on Higgs Mass
top
• Running Higgs self-coupling:
Higgs
+
=-
DM fluctuations
+
without DM
with DM
11
RG flow
⇤EW
> 0: lower Higgs mass
⇤UV
• Contours of fixed cutoff scale:
12
‣ SM:
500
SUV =
Z
1
1
d4 x ¯i@/ + (@µ h)2 + (@µ S)2 + iȳh ¯ + V̄ (h, S)
2
2
M h (GeV )
400
Λ=1019 GeV
Λ=103 GeV
300
200
Λ=106 GeV
Λ=109 GeV
100
0
0.0
0.2
0.4
)/2)
11 (M
a Z(M
2
Z
0.6
0.8
1.0
Gonderinger et al. (2009)
Eichhorn, MMS (2014)
Standard Model as a Low-Energy Effective Theory
E↵ective field theory and higher-order couplings
• Potential at UV scale: all operators compatible with symmetries
induced potential at ⇤: all operators compatible with symmetries!
¯6 6
¯8 8
4 4
(⇤)
4
6 (⇤)i = 6¯ ⇤i 4 ⇠ O(1)
V⇤ = 4
+ 8
+ 16 +
4 ... with
VUV =
4
H +
8⇤2
H + ...
in the IR: canonical scaling:
• Towards IR: irrelevant operators follow canonical scaling
Fodor et al. (2008)
Branchina & Messina (2013)
λ6
3.0
Gies et al. (2013)
becomes tiny very fast!
2.5
2.0
on0mass
bounds
‣ Nevertheless:
assumption:impact
for i >
4:
i (⇤) =
1.5
on maximal UV extension
‣ Or: impact
m
1.0
H
0.5
100.00 106 1010 1014 1018
k/GeV
200
Ê
150Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
100
becomes tiny very fast
) negligible for Mh (⇤)?
No. [IR observable?]
suggested in [Fodor, Holland, Kuti, Nogradi, Schroeder, 2008]
RG [Branchina, Messina, 2013; Gies, Gneiting, Sondenheimer, 2013]
50
8
10
12
14
16
18
Log@10, LêGeVD
Standard Model as a Low-Energy Effective Theory
E↵ective field theory and higher-order couplings
• Potential at UV scale: all operators compatible with symmetries
induced potential at ⇤: all operators compatible with symmetries!
¯6 6
¯8 8
4 4
(⇤)
4
6 (⇤)i = 6¯ ⇤i 4 ⇠ O(1)
V⇤ = 4
+ 8
+ 16 +
4 ... with
VUV =
4
H +
8⇤2
H + ...
in the IR: canonical scaling:
• Towards IR: irrelevant operators follow canonical scaling
Branchina & Messina (2013)
λ6
3.0
Gies et al. (2013)
becomes tiny very fast!
2.5
2.0
on0mass
bounds
‣ Nevertheless:
assumption:impact
for i >
4:
i (⇤) =
1.5
on maximal UV extension
‣ Or: impact
m
1.0
H
0.5
6
10
14
18
100.00
10
10
10
10
Main idea
k/GeV
200
Ê
150Ê
• Mechanism to go below lower mass bound:
becomes tiny very fast
0.12
λ4)
) negligible
for M
h0(⇤)?
1. Choose Higgs-self-coupling
<
at UV scale
new)physics)induces))
0.10
generalized)Higgs)poten9al)
No.
[IR
observable?]
0.08
6
2. Choose Φ -coupling
> 0Holland,
→ potential
stable
suggested in [Fodor,
Kuti, Nogradi,
Schroeder,
2008]
λ6) is
0.06
running couplings
Fodor et al. (2008)
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
VUV
100
50
8
10
12
14
Log@10,
LêGeVD
6
without Φ -coupling
16
18
VUV
0.04
RG [Branchina, Messina, 2013; Gies, Gneiting, Sondenheimer, 2013]
➡ Obtain smaller
Higgs-self-coupling in the IR
0.02
0.00
➡ Higgs mass
-0.02 lower than lower bound!
5
9
13
17
10
10
10
Λ) 10
RG scale in GeV
10
with Φ6-coupling
Functional RG
•
Nonperturbative tool: functional RG
Use functional RG method as an appropriate tool to obtain β functions:
@t
k
1
=
Tr @t Rk (
2
⇤ RG trajectory: operators
‣ allows to include all quantum fluctuations in presence of higher-dim
k !0
(2)
k
+ Rk )
=
=
‣ flowing action Γk with RG scale k interpolates between
microscopic action (k ! ⇤) :
full e↵ective action (k ! 0) :
‣ FRG flow equation:
1
@t k [ ] = STr{[
2
(2)
k [
] + Rk ]
k[
k[
1
] ! S[ ]
]! [ ]
(@t Rk )} .
Wetterich (1993)
β functions for model couplings…
…(reproduce 1-loop β functions from PT, include threshold effects, higher-dim operators,…)
R
1
Ueff + . . .
already observe a similar perturbative flow toward negative 4 at high scales [9]. The running of the top Yuk
coupling to smaller values in the ultraviolet is included when we add an SU (3) gauge sector. Correspondingly
Gauged
Higgs-Top
Model
- Higher-dimensional
operators
investigate
the Euclidean
action defined
at the UV-cuto↵
scale ⇤
2
3
nf
Z
ny
X
1
y X
2
4 41
µ⌫
/ j + ip
S⇤ = d x
Fµ⌫ F + (@µ ') + Ve↵ (⇤) + i
' j j5 ,
jD
4
2
2 j=1
j=1
• Potential at UV scale: completely stable with unique minimum at H=0
running couplings
running couplings
with an SU (Nc ) gauge field (Nc = 3), a real scalar ', Dirac fermions j for nf flavors, the covariant derivative
including the strong coupling gs , and the e↵ective
potential
4 (⇤) scalar
6 (⇤) 6 Ve↵ . We assume that ny < nf of the ferm
4
VUV =
H Yukawa
+
H + ...y. The remaining n
2
species are heavy and shall have a degenerate
large
coupling
ny flavors
f
4
8⇤
negligible
Yukawa couplings. The e↵ective scalar potential1.0is now expanded in terms of dimensionless coupl
1.0
2n as
0.058
0.05
4 (⇤) =
4 (⇤) =
g3
g3
✓ 2 ◆n
0.8
0.8
2
X
(⇤)
=
2.0
y
6
6 (⇤) = 0.8
µ(k) 2
2n (k) y '
Ve↵ (k) =
' +
.
2n 4
2
k
2
0.6
0.6
n=2
0.4 potential should be compared to the Standard Model0.4
This
potential shown in Eq.(1). As part of the potentia
study beta functions for the self-interaction 4 and the '6 -coupling 6 . Here we restrict ourselves to the fir
0.2
an 0.2
(infinite) series of possible higher-order couplings.
l4
l4
0.0 l6 our toy model does not reflect the weak gauge 0.0
l6
Because
structure
of the Standard Model, the gauge coupl
g1 and
not appear.
However,
we know
that in
the Standard
Model they
have11 significant
e↵ects:
first,
100 g2 do
100
108
1011
1014
1017
1020
108
10
1014
1017
1020
105
105
gauge couplings give a significant positive contribution to 4 , balancing the negative top Yukawa terms for s
k in GeV
RGE scale k in GeV
values of 4 ; second, RGE
theyscale
slow
down the growing top Yukawa and thereby a↵ect
the increase of 4 toward l
field values. Since the variation of the weak coupling at large energy scales is modest, we account for its e↵
ure by
4: including
Running couplings
including in
Eq.(7)
with mh = 125
The
bare pote
4 and
6 , corresponding
a finite contribution
the beta
functions for to
y, parametrized
by aGeV.
fiducial
coupling
gF
4 and
sfy the stability condition of Eq.(14) and thus remain bound from below at all lower scales. Left: 4 (⇤) = 0.05
⇤) = 2.0 and ⇤ = 5 · 1011 GeV. Right: 4 (⇤) = 0.050 and 6 (⇤) = 0.8 and ⇤ = 1019 GeV.
rators. In between these two scales an e↵ective field theory description within the model is still possible,
already observe a similar perturbative flow toward negative 4 at high scales [9]. The running of the top Yuk
coupling to smaller values in the ultraviolet is included when we add an SU (3) gauge sector. Correspondingly
Gauged
Higgs-Top
Model
- Higher-dimensional
operators
investigate
the Euclidean
action defined
at the UV-cuto↵
scale ⇤
2
3
nf
Z
ny
X
1
y X
2
4 41
µ⌫
/ j + ip
S⇤ = d x
Fµ⌫ F + (@µ ') + Ve↵ (⇤) + i
' j j5 ,
jD
4
2
2 j=1
j=1
• Potential at UV scale: completely stable with unique minimum at H=0
running couplings
running couplings
with an SU (Nc ) gauge field (Nc = 3), a real scalar ', Dirac fermions j for nf flavors, the covariant derivative
including the strong coupling gs , and the e↵ective
potential
4 (⇤) scalar
6 (⇤) 6 Ve↵ . We assume that ny < nf of the ferm
4
VUV =
H Yukawa
+
H + ...y. The remaining n
2
species are heavy and shall have a degenerate
large
coupling
ny flavors
f
4
8⇤
negligible
Yukawa couplings. The e↵ective scalar potential1.0is now expanded in terms of dimensionless coupl
1.0
2n as
0.058
0.05
4 (⇤) =
4 (⇤) =
g3
g3
✓ 2 ◆n
0.8
0.8
2
X
(⇤)
=
2.0
y
6
6 (⇤) = 0.8
µ(k) 2
2n (k) y '
Ve↵ (k) =
' +
.
2n 4
2
k
2
0.6
0.6
n=2
0.4 potential should be compared to the Standard Model0.4
This
potential shown in Eq.(1). As part of the potentia
study beta functions for the self-interaction 4 and the '6 -coupling 6 . Here we restrict ourselves to the fir
0.2
an 0.2
(infinite) series of possible higher-order couplings.
l4
l4
0.0 l6 our toy model does not reflect the weak gauge 0.0
l6
Because
structure
of the Standard Model, the gauge coupl
g1 and
not appear.
However,
we know
that in
the Standard
Model they
have11 significant
e↵ects:
first,
100 g2 do
100
108
1011
1014
1017
1020
108
10
1014
1017
1020
105
105
gauge couplings give a significant positive contribution to 4 , balancing the negative top Yukawa terms for s
k in GeV
RGE scale k in GeV
values of 4 ; second, RGE
theyscale
slow
down the growing top Yukawa and thereby a↵ect
the increase of 4 toward l
field values. Since the variation of the weak coupling at large energy scales is modest, we account for its e↵
ure by
4: including
Running couplings
including in
Eq.(7)
with mh = 125
The
bare pote
4 and
6 , corresponding
a finite contribution
the beta
functions for to
y, parametrized
by aGeV.
fiducial
coupling
gF
4 and
Potential
completely
stable
during
entire
RG
flow
‣
sfy the stability condition of Eq.(14) and thus remain bound from below at all lower scales. Left: 4 (⇤) = 0.05
11
⇤) =
2.0 andUV
⇤=
5 · 10by
GeV. of
Right:
= 2) 0.050 and 6 (⇤) = 0.8 and ⇤ = 1019 GeV.
cutoff
orders
magnitude
‣ Extend
4 (⇤)(~
rators. In between these two scales an e↵ective field theory description within the model is still possible,
already observe a similar perturbative flow toward negative 4 at high scales [9]. The running of the top Yuk
coupling to smaller values in the ultraviolet is included when we add an SU (3) gauge sector. Correspondingly
Gauged
Higgs-Top
Model
- Higher-dimensional
operators
investigate
the Euclidean
action defined
at the UV-cuto↵
scale ⇤
2
3
nf
Z
ny
X
1
y X
2
4 41
µ⌫
/ j + ip
S⇤ = d x
Fµ⌫ F + (@µ ') + Ve↵ (⇤) + i
' j j5 ,
jD
4
2
2 j=1
j=1
• Potential at UV scale: completely stable with unique minimum at H=0
running couplings
running couplings
with an SU (Nc ) gauge field (Nc = 3), a real scalar ', Dirac fermions j for nf flavors, the covariant derivative
including the strong coupling gs , and the e↵ective
potential
4 (⇤) scalar
6 (⇤) 6 Ve↵ . We assume that ny < nf of the ferm
4
VUV =
H Yukawa
+
H + ...y. The remaining n
2
species are heavy and shall have a degenerate
large
coupling
ny flavors
f
4
8⇤
negligible
Yukawa couplings. The e↵ective scalar potential1.0is now expanded in terms of dimensionless coupl
1.0
2n as
0.058
0.05
4 (⇤) =
4 (⇤) =
g3
g3
✓ 2 ◆n
0.8
0.8
2
X
(⇤)
=
2.0
y
6
6 (⇤) = 0.8
µ(k) 2
2n (k) y '
Ve↵ (k) =
' +
.
2n 4
2
k
2
0.6
0.6
n=2
0.4 potential should be compared to the Standard Model0.4
This
potential shown in Eq.(1). As part of the potentia
study beta functions for the self-interaction 4 and the '6 -coupling 6 . Here we restrict ourselves to the fir
0.2
an 0.2
(infinite) series of possible higher-order couplings.
l4
l4
0.0 l6 our toy model does not reflect the weak gauge 0.0
l6
Because
structure
of the Standard Model, the gauge coupl
g1 and
not appear.
However,
we know
that in
the Standard
Model they
have11 significant
e↵ects:
first,
100 g2 do
100
108
1011
1014
1017
1020
108
10
1014
1017
1020
105
105
gauge couplings give a significant positive contribution to 4 , balancing the negative top Yukawa terms for s
k in GeV
RGE scale k in GeV
values of 4 ; second, RGE
theyscale
slow
down the growing top Yukawa and thereby a↵ect
the increase of 4 toward l
field values. Since the variation of the weak coupling at large energy scales is modest, we account for its e↵
ure by
4: including
Running couplings
including in
Eq.(7)
with mh = 125
The
bare pote
4 and
6 , corresponding
a finite contribution
the beta
functions for to
y, parametrized
by aGeV.
fiducial
coupling
gF
4 and
Potential
develops
2nd
Minimum
during
RG
flow
Potential
completely
stable
during
entire
RG
flow
‣
‣
sfy the stability condition of Eq.(14) and thus remain bound from below at all lower scales. Left: 4 (⇤) = 0.05
11
⇤) =
2.0 andUV
⇤=
5 · 10by
GeV. of
Right:
= 2) 0.050 and 6 (⇤) ‣= Min
0.8 and
⇤ =only
1019
GeV.
@ H=0
metastable
cutoff
orders
magnitude
‣ Extend
4 (⇤)(~
‣ Small λ6 sufficient to stabilize UV potential
studies
required…
‣ Further
rators. In between these two scales an e↵ective field theory
description
within
the model is still possible,
mh (MPl ) ⇡ 129GeV
mexp
⇡ 125GeV
h
Λ for observed Higgs mass
135
l4 HLL=0
conventional mass bound ~ 129 GeV
l4 HLL=-
130
125
l4 HLL=0, l6 HLL=0.5
115
l4 HLL=-0.01, l6 HLL=0.05
l4 HLL=-0.05, l6 HLL=0.8
105
1010
1012 1014
LêGeV
1016
l4 HLL=-
0 metastable phase
dip into possible
l4 HLL=0, l6 HLL=2
110
10
relax mass5bounds/increase UV cutoff
l4 HLL=0, l6 HLL=0
120
108
l4 HLL=0
15
Dmh in %
mh êGeV
Mass Bounds with Higher-Dimensional Operators
1018
-5
-10
10
generalized UV
8 potentials
10
10
1012 1014
LêGeV
re 5: Infrared value of the Higgs mass (left) and the relative shift mh measuring the departure fr
shifts at
of 1-5%
viable ⇤. We show di↵erent UV boundary conditions as i
‣ (right),
r bound
as level
a function
of seem
the UV-cuto↵
Stability Regions
Ê
1019
L@GeVD
1017
1015
10
13
Ê
¨III
Vint êk
4
mh =
V êk125V GeV
êk
k
4
VIR êk 4
UV
VUV êk 4
VIR êk 4
Vint êk 4
Ê
Ê
Vk êk 4
4
Hêk
II
Hêk
Vk êk 4
VUV êk 4
V êk
Figure 3. Starting from a stable potential in the UV
di↵erent flows
V êk to the IR
Ê minimum areÊpossible. In theÊ left panel we show a situation where
the electroweak
Ê
Ê
11 ÊÊ Ê Ê
remains stable during the whole flow from k = ⇤ to the point where the electrow
10 Ê
forms. This corresponds to region I in Fig. 6. For smaller initial values of 6 cor
Hêk
Ê
region II in Fig. 6, we observe a meta-stable behavior of the scale dependent poten
9
10
intermediate values of k, which then disappears. The electroweak symmetry bre
0.0
0.5 are not formed
1.0 until later. 1.5
2.0
We call this behavior pseudo-stable.
Figure 3. Starting from a stable potential in th
l6 HLL
the electroweak minimum are possible. In the left
Eq.(2.14) is violated first, a second
appears
and flow
the minimum
remainsminimum
stable during
the whole
from k = ⇤a
of magnitude
at full to
stability
(green)
‣ Moderately small λ6(Λ) extend UV cutoff by 2 orders
corresponds
region
I in Fig.
Fo
become meta-stable for someforms.
rangeThis
of RG
scales, cf.
right panel
in 6.Fig.
region
IIofinthe
Fig.full
6, we
observe awill
meta-stable
- allows
for more orders
‣ Pseudo-stable region (blue)
a reliable
(meta-)stability
analysis
potential
require behav
to go
6of magnitude
intermediate values of k, which then disappears.
simple
polynomial
expansion
of the e↵ective potential around one minimu
➡ extend FRG study
are not formed until later. We call this behavior
int
4
IR
4
I
ability regions as a function of ⇤ and . In region I (green) the pot
n region II (blue) the For
UV-potential
is
stable,
while
the
potential
i
completeness let us note what happens when we continue to
➡ possibility
effectiveIII
potential
at k the
≈ 0 behavior
mediate scales
v <energy
k of<meta-stable
⇤; inIndependently
region
(red)
UV-potential
vi
scales.
of the
high-scale
discussed above,
Eq.(2.14) is violated first, a second minimum
Higgs Mass (Bounds) with Light Dark-Matter Scalar
• toy model:
SUV =
Z
1
1
d4 x ¯i@/ + (@µ h)2 + (@µ S)2 + iȳh ¯ + V̄ (h, S)
2
2
• Fix vev = 246 GeV and mtop = 173 GeV
• Choose
4
=
0.05 and
6
= 0.5
➡ Higgs masses with dark matter and new couplings:
‣ toy model!
vanishing higher order couplings
mH
200
Ê
Ê
Ê
Ê
150
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
with dark matter
Ê
Ê
8
10
12
mass with Φ6 and DM
Log@10, LêG
14
16
18
Log10 (⇤/GeV)
Eichhorn, MMS (2014)
Conclusion
•
measured Higgs mass very close to lower bound mh(Λ = MPl)
Ê
19
•
perturbative analysis: Higgs potential loses stability around 1010 GeV
L@GeVD
•
10
this statement can be relaxed:
1017
Ê
¨III
Ê
1015
13
10
1011 ÊÊÊ
‣ higher-dimensional operators at UV scale Λ
II
Ê
Ê
Ê
Ê
Ê
1.5
2.0
I
Ê
109
0.0
Ê
Ê
Ê
0.5
l6 HLL
1.0
‣ non-perturbative treatment allows for more general values of higher-dim couplings
✓
Higgs masses below lower bound are possible
Figure 6. Di↵erent stability regions as a function of ⇤ and 6 . In region
is stable everywhere; in region II (blue) the UV-potential is stable, whil
pseudo-stable for intermediate scales v < k < ⇤; in region III (red) the
the unique-minimum condition. Our simple model includes the e↵ects of
bosons only in the running of 4 . In Appendix E, we model the electro
the full Higgs potential. The corresponding region III is then somewhat
So far, we have restricted ourselves to absolutely stable ba
minimum at vanishing field values. Next, we turn to bare pote
below, but showing two minima. Presumably, this property also h
potential. However, since we employ a polynomial expansion of
only study the renormalization group flow around
Pl the electrowea
Future studies based on a di↵erent expansion will have to shed
global renormalization group flow of the e↵ective potential.
As already discussed in Sect. 2.3, even a very small positive va
to ensure stability. However, the point H = 0 is typically only
case. The phenomenological viability of this scenario requires tha
the minima is sufficiently slow. To ensure this we enforce the con
App. B),
8k.
4 (k) > 0.052,
✓
with completely stable potential, we can extend UV cutoff by 2 orders of magnitude
✓
DM fluctuations allow for smaller Higgs masses at fixed Λ (~ a few GeV @ Λ = M )
✤
Question: What type of physics can predict higher-dim operators of suitable size?
‣ we have investigated simple SM extension with heavy scalars
With this choice, the limit therefore essentially reduces to the usu
the meta-stable region as obtained in the absence of any 6 .
Our numerical study based on a more advanced approxima
App. C shows that already with moderately small values 6 ⇡
JHEP
04 (2015)
022 and arXiv:1501.02812
can be
increased
by approximately
two orders of magnitude wh
stability
of the
vacuum.
be seen from the gre
Phys.
Rev.electroweak
D 90 (2014)
025023This
andcan
arXiv:1404.5962
Increasing the possible cuto↵ scales by further orders of magnit
‣ required parameter choices in simple model are at border to non-perturbative