Higgs Mass and the Scale of New Physics

Higgs Mass 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
May 4, 2015 @ MPIK, Heidelberg
JHEP 04 (2015) 022 or arXiv:1501.02812
Phys. Rev. D 90 (2014) 025023 or arXiv:1404.5962
Outline
•
Introduction
‣ Higgs Mass Bounds
‣ Vacuum Instability
‣ The Scale of New Physics
•
Standard Model as Low-Energy Effective Field Theory
‣ Gauged Higgs-Top Model
‣ Impact of Higher-Dimensional Operators
‣ Generation of Higher-Dimensional Operators
•
Conclusions
[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
Standard Model Running Couplings
matching at the heavy quark pole masses Mc = 1.5 GeV and Mb = 4.7 GeV. Results from
ges of energies are only given for Q = MZ0 . Where available, the table also contains the
ns of experimental and theoretical uncertainties to the total errors in αs (MZ0 ).
in the last two columns of table 1, the underlying theoretical calculation for each meand a reference to this result are given, where NLO stands for next-to-leading order, NNLO
xt-to-leading-order of perturbation theory, “resum” stands for resummend NLO calculations
de NLO plus resummation of all leading und next-to-leading logarithms to all orders (see
S
2]), and “LGT” indicates lattice gauge theory.
‣ Running strong coupling α :
‣ Standard model running couplings:
Bethke (2007)
1.0
SM couplings
0.8
Buttazzo et al. (2013)
g3
yt
g2
0.6
g1
0.4
0.2
0.0
102
Energy scale
yb
m in TeV
l
104
106
108
1010
1012
1014
1016
1018
1020
RGE scale m in GeV
. Summary of measurements of αs (Q) as a function of the respective energy scale Q, from
en symbols indicate (resummed) NLO, and filled symbols NNLO QCD calculations used in
ve analysis. The curves are the QCD predictions for the combined world average value of
4-loop approximation and using 3-loop threshold matching at the heavy quark pole masses
eV and Mb = 4.7 GeV.
t
b
1.0
Figure 1: Renormalisation
of the SM gauge couplings g1 = 5/3gY , g2 , g3 , of
g3 Higgs quartic coupling and of the Higgs ma
and couplings (y , y , y ), ofy the
0.8
e 17, all results of α (Q) given in table 1 are graphically
displayed, as a function
the
All parameters
are ofdefined
in
the ms scheme.group
We include
two-loop thresholds a
renormalization
β functions
e Q. Those results obtained in ranges of Q and given, in table 1, as α (M ) only, are not
this figure - with one exception: the results fromand
jet production
in deep inelastic
scattering
three-loop
RG
equations. The thickness indicates the ±1 uncertainties in
s
Z0
plings
s
0.6
nted in table 1 by one line, averaging over a range in Q from 6 to 100 GeV, while in figure 17
esults for fixed values of Q as presented in [67] are displayed.
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
top loop
kEW
RG flow
= 0 at ⇤UV :
➡ minimal value of Higgs mass
⇤UV RG scale
4
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
5
=
=
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 21.0
2
1.0
=
y Buttazzo
8gset al. (2013)
g2
g1 y
y =
2
dyt logg3k
16⇡ 2
4
12• Vacuum instability
0.8

0.8
µ 2
3
4
4
d gs
gs
2
H
+
H
crosses
zero
in
‣
4
2
4
=
=
11
n
.
0.6
gs 0.6
f
g2
2
d log k
16⇡
3
➡ instability of Higgs vacuum
g1
0.4
p
0.4
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
‣0.2
H
tributions0.2 from dimension-6m inoperators.
[Typically, these
TeV
l4 coupling parameters or the related
l
0.0
g3
mt and m
,
are
evaluated at the scale of the top pole
mass to be translated
into the pole m
H
y
b
0.0
FIG. 3. Height of the potential barr
gs and the 10
top,
according to the on-shell scheme.] The
two gauge
couplings
are
g2 10for
SU
(2)L a
8
17
2
100
10
1011
1014 Yukawa:
1020
104 106 108 1010 1012 1014 1016 1018 1020
105 depends
strongly
on
top
‣
2
(1)Y . The number of fermions
contributing
to
the running of the strong
is nf . In the
RG scale k incoupling
GeV
RGE scale m in GeV
[
1e+18
o We
explicit higher
dimensional operators
are present. However, if they are generated by Standar[
1.2
Gabrielli et al. (2013)
Bezrukov et al. (2014)
0.6
g2
g1
mass, we find 0.4
the following
values of the SM parameters:
0.4
r are
g1 (MPl ) =0.20.6168
run0.2
g2 (MPl ) = 0.5057
alidll 4
0.0
0.0
ysics
0.1184
3 (MZ )
g3 (MPl ) = 0.4873
5 + 0.0002
10
15
20
25
per0.0007
✓108 log1010HmêGeVL
◆1014
11
100
105
ticle
Mt
y (M ) = 0.3823 + 0.0051
173.35
µnew, GeV
0.6
running couplings
ment
runningSM
couplings
Couplings
1.0
1:
Renormalisation
of the SM gauge couplings g1 = 5/3gY , g2 , g3 , of the top,1.0bottom
1e+16
d to
g
3
ouplings
(yt , yb1.0
, y ), ofy the Higgs quartic coupling and of the Higgs mass parameter m. g3
new
0.8
0.8 1e+14
g3 the ms scheme. We include two-loop thresholds at the weak scale
y
ameters
are
defined
in
ctor.
0.8
ee-loop
uncertainties in Mt , Mh , 3 . 1e+12
yt The thickness indicates the ±1
atu- RG equations.
0.6
0.4
1e+10
0.2
1e+08
(56a)
(56b)
1e+06
l
0.0
30
17
10
0.0021
35
40
3 (MZ )
20
10
0.1184
4
0
(56c)
100
(56d)
10
5
0.01
10
8
0.02
0.03
0.04
0.05
1017
1020
yt-ytcrit11(µ=173.2
GeV)
14
10
10
i
a
O
b
o
c
t
o
n
i
i
h
e
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:
V (k = ⇤; H) = VUV
V (k = 0; H) = Ve↵ .
1. New degrees of freedom appear that In
render
Higgsthepotential
stableproperties
- dark matter?
this manner,
(meta-)stability
of the e↵ective potenti
in a scale-dependent manner.
1040
Gabrielli et al. (2013)
note another subtlety in the
comparison between
-10
and non-perturbative -10
approaches. The perturbative approach usually
-10
-10
True minimum @ H ~ 1025 GeV?
independent regularization
scheme. To avoid large threshold e↵ects on
-10
stop the running of the
couplings at the appropriate mass scales. Mo
-10
to H. This suggests to approximate
Metastability of Higgs vacuum?masses are proportional
2. Stable minimum might appear for large field
values
Finally,
let us
➡
➡
40
50
60
70
VHhLêGeV4
➡
Small tunnelling rates to stable minimum?
80
90
-10100
Ve↵
1
= V (k = H, H) ⇡
4
4 (H) H
4
,
5
10
15
20
25
2
as is usually
done in the perturbative
approach.
In Eq.(2.12)
we have
log HhêGeVL
the running Higgs self-coupling at the scale kEW . Using the identifi
Gabriellifield
et al. (2013)
FIG.to
2: The
SMterm
Higgs
potential
as a function
the
essentially
the
first
ine↵ective
the square
brackets
of theofsecond
Higgs
(e.g.
~H6,reproduce
H8, …)
render
potential
stable
2 2
4
Higgs field strength, V (h) = µ h + H (h) h . The Higgs
Let us check thismass
explicitly,
parameter is approximated as a constant and the run10
1.2
Include
higher powers in
1.0
➡
SM Couplings
We 3.
d to
new
ctor.
atument
r are
runalidysics
perticle
and
0.8
ning quartic
coupling
is evaluated at 1-loop level, where the
g
Do
not appear in perturbatively renormalizable
Higgs
Lagrangian
V (k ) ⇡ V scale
+ is Vset by the field strength h. The global minimum at
3
e↵
yt
➡
0.6
EW
g
Appear
in effective theories with finite Λ
2
g1
0.4
➡
0.2
➡
0.0
top
UV
26
✓ positive 2(from
◆ low
⇠
10
GeV
is generated by
H running
h
2
2
µ (⇤) c2 ⇤ 2at this 1scale. The
2⇤at the elec- 2
4 4local minimum
UV to high energy)
=
H
+
y
H
log
1
+
+ 2y ⇤
2
troweak
by2the negative Higgs mass
parameter.
2 scale is caused
64⇡
y2H
New physics appears at higher scales
l
Link
to
particle
models?
5
10 BSM
15
20
25 physics
30
35
40
log10 HmêGeVL
when approaching underlying theory
✓
◆
2H 2 i
y
4 (⇤) 4
10? GeV
> 10110+ GeV
4⇤4 log
+
H
2
2⇤
4
✓
◆
toward
negative
values
(from
high
to low2 scale).
Howh
2
µ (kEW ) 2SM contains
1
2⇤ with its large
4also
4 the top quark
2 2 2
= ever, the
H +
y
H
log
1
+
+
2y
⇤ H
2 the Higgs boson.y 2As
Yukawa
coupling
2
64⇡to
H 2 explained in
✓
◆i
the last section,
at2 low
near the EW scale the
2 energies
y
H
4 (⇤) 4
4
Stability & Higher-Dimensional Operators
approximate weak-scale potential
briefly
recalling
issueinduce
of vacuum
stability
arises
and
what
Does
thehow
topthe
loop
an
instability
in
the
potential?
µ2 (⇤) c2 ⇤2 2
(⇤)
4
4
)
⇡
V
+
V
=
H
+
H
+ positive terms, (2.1
lays. We refer toVe↵a(kpotential
as
stable
if
it
is
bounded
from
below,
top
EW
UV
2
4
alone,
but
requires
the
full
UV-potential.
As
simple example, we includ
able. A stable potential can exhibit several minima, where aa local
and will in
never
toofan
unstable
e↵ective
a finite
UV-cutoff
Λ: potential at the weak scale. A closer loo
• Vacuum stability
aspresence
in lead
Eq.(2.1),
meta-stable while
global minimum
is stable.
Asany
argued
induce
in ⇤the
presence
a fi
at thethe
contribution
fromand
thecannot
top
loop
below
theinstability
cuto↵above,
scale
reveals
the of
formal
2 (⇤)
µ
4 (⇤)
6 (⇤)
ar the ➡dominant
factor
in
the
renormalization
group
evolution
of
2
4
6
To
understand
this
in
more
detail,
let
us
consider
a o
sub-leading
terms
H +
H +
H
.
start with stable bare potential VUV = Ve↵ (⇤) =
2
2
4
8⇤
fective
potential
the2 understood
electroweak scale k ⇡ kEW ⇡ 0
✓ at 2be
◆
. Indeed, the whole issue of stability
already
Z ⇤ can
1
y H
3is essentially
As
we
have
seen,
(⇤)
by the
measured
value
of the de
Hi
4
2
when
appropriate.
shown
in(2.7).
Ref.
[15] the
fermion
4
= /(8⇡
dq· q· · log
1 + Asfixed
➡ the
consider:
nsidering
term 4 =Vtop3y
)
+
in
Eqs.(2.2)
and
2
2
4⇡ 0
2q
2 (⇤) is fixed
the infrared. For
sufficiently
large
⇤
it
is
negative.
Similarly
µ
contribution tohthe e↵ective
✓ potential
◆ for any finite cuto
2
2
unction toward the ultraviolet,
this
term
drives
the
quartic
self1 For
2⇤
2 infrared.
2
4 the
4
2 2 2cuto↵ reg
v = 246=GeVy in ⇤the
simple
momentum-space
H
+
y
H
log
1
+
+
2y
⇤ H
2
2
2
2
16⇡
64⇡
y =
HΛ):
2
2scale
2 . As2 expected,
e values
at some
highit scale.
This
ispotential
the
usual
argument
for
the
➡ top
loop contribution
to effective
@
EW
(k
<<
is typically
large
and
positive,
µ
(⇤)
⇠
O(0.01)⇤
V
c
⇤
H
+ positive
t
EW
6 (⇤)
top
2
✓
◆
i
2H 2
y
4
4
he Higgs potential if the
by undetermined
V ⇡
new Higgs
and freepotential
parameter.
Its
value is essentially
by measurem
4⇤ log 1is+
yapproximated
⇤
2 2
42
2⇤
V
=
c
⇤
H
+c
H
log
+ ... scale
top because
2 it is csuppressed
4 0. We
where
start
from
a stable
bare potential VU
2 >
infrared,
by
the
large
⇤.
4
k
EW
4
|
{z
} UV-potential
y
⇤
in
Eq.(2.1)
with
quadratic
and
quartic Higgs
terms.
T
26 (⇤)
2 positive
4ensures
Choosing
that
the
grows
at
large
=
c
⇤
H
+
c
H
log
+
.
.
.
.
(2.1
positive terms
4
n appears to be in conflict with2 non-perturbative
lattice simula4
k
approximate
weak-scale
potentialfield theory. In principle,
making it a viable
bare potential
forEW
a quantum
➡ Higgs
@ EW scale:
arguments
thatpotential
the interacting
part
of
the topsmall
loopvalue
is non-negative
2 (⇤) > 0, the po
achieved
with
an
arbitrarily
of
(⇤).
Since
µ
6
where we normalize the contribution such that Vtop (H = 20) = 0. In the
last line w
2
µ
(⇤)
c
⇤
2panel 2of Fig.
4 (⇤
has
a
qualitatively
similar
form
to
the
red
curve
in
the
left
(kEW ) ⇡The
VUVadditional
+ Vtop =
H + to 2th
adopt the general notation V
ofe↵Eq.(2.9).
logarithmic contribution
2 value H 6= 0. 4Ch
minimum
at
H
=
0
and
a
global
minimum
at
a
large
field
quartic potential term contributes to the Higgs mass as [14, 22]
and will never lead to an unstable
2 e↵ective potential at
⇤
1
2⇤
4
(⇤)
>
(⇤)
= 4 (⇤) +6c4 yfrom
log 4the top
⇡ 2 (⇤)
.
(2.1
➡ contribution to–quartic
loop
below
the GeV
cuto↵
MH ≈ 125
4 (kthe
EW )contribution
8 – term: at
3µ
kEW
8
sub-leading terms
removes
thethe
second
minimum atofHthe
6= 0Higgs
for negative
only
minim
4 (⇤). The
This
is
nothing
but
renormalization
self-coupling.
For
a
sufficient
➡ Large Λ forces us to choose λ4(Λ) < 0 to obtain measured
✓
◆
Z ⇤Higgs mass!
2
2
atscale
H = ⇤0,the
as shown
by the
blue curveforces
in1 Fig.
we
are
mainl
y H
large cuto↵
above top
contribution
us 2.
to In
choose
a negative
high-sca
3this paper
Vtop =
dq q log 1 +
minimum. In the course of the RG flow, al
Stabilizing the UV potential
from below shown in Fig. 3 do occur, depe
• No instability induced by top loop!
Qualitatively, this behavior can be un
es the full UV-potential. As a simple example, we include a 6 term
• However: have to chose unstable Φ potential @ Λ to reproduce M = 125 GeV
in Eq.(2.14), which can be applied to an
µ2 (⇤)
Idea:
include
higher-dimensional
operators!
4 (⇤) 4
6 (⇤) 6
•
2full UV-potential.
for
decreasing
k,
grows
the
same
ti
alone,
but
requires
the
simple
we include at
a
term
VUV = Ve↵ (⇤) =
H +
H + As2 a H
. example,
(2.13)
4
2
4
8⇤
as in Eq.(2.1),
fixed by vev = 246 GeV
free
turns
positive
then
the
scale
depen
by the measured
value of first,
the Higgs mass
in parameter
4 (⇤) is essentially fixed
µ (⇤)
(⇤)
(⇤)
2
= V (⇤) =
H µ+(⇤) is fixed
H +by requiring
H .
(2.13)
sufficiently large ⇤ itV is negative.
Similarly
2
4
8⇤
RGmomentum-space
evolutioncuto↵
infixed
our
approximation,
cf.
the infrared. For the simple
regularization,
by Higgs mass
4
UV
H
6
2
UV
2
e↵
4
4
6
6
2
As we have seen,
fixed by the measured value of the Higgs mass in
4 (⇤) is essentially
2
2
ge and positive, µ (⇤) ⇠ O(0.01)⇤ . As expected, 6 (⇤) is 2the only
the infrared. For sufficiently large ⇤ it is negative. Similarly µ (⇤) is fixed by requiring
ameter. Its value is essentially undetermined by measurements in the
v = 246 GeV in the infrared. For the simple momentum-space cuto↵ regularization,
it isitsuppressed
theand
large
scale ⇤.
is typically by
large
positive,
µ2 (⇤) ⇠ O(0.01)⇤2 . As expected, 6 (⇤) isVthe
UV only
(⇤) positive
that thepotential
UV-potential
growsundetermined
at large fieldbyvalues,
stable
new
and ensures
freeλparameter.
value is is
essentially
measurements in the
6(Λ) → UVIts
• positive
le bare
potential
for it
a is
quantum
field
principle,
this can be
infrared,
because
suppressed
bytheory.
the largeInscale
⇤.
2 (⇤) > 0, the potential then
arbitrarily
smallndvalue
of
(⇤).
Since
µ
Choosing
(⇤)
positive
ensures
that
6
• remove 2 6minimum for λ4(Λ) < 0: the UV-potential grows at large field values,
ly similar
the bare
red curve
in the
panel offield
Fig.theory.
2 withInaprinciple,
local
makingform
it a to
viable
potential
for left
a quantum
this can be
achieved
withminimum
an arbitrarily
Since
µ2 (⇤) > 0, the potential then
= 0 and
a global
at a small
largevalue
field of
value
H 6=
0. Choosing
6 (⇤).
has a qualitatively similar form to
the red curve in the left panel of Fig. 2 with a local
2
⇤ minimum at a large field value H 6= 0. Choosing
2 a global
minimum at H
=
0
and
(⇤)
(2.14)
6 (⇤) >
4
3µ2 (⇤)
6 (⇤)
>
2
4 (⇤)
⇤2
2
(2.14)
H
The perturbative approach starts from the usual Higgs potential, generalized to an e↵ective potential
by allowing
a dependence
on the momentum
scale k of all parameters.
RG
Flowforof
Generalized
Potentials
Including higher-dimensional operators it reads
✓ 2 ◆n
2
X
µ(k) 2
H
µ2 (k) 2
2n (k)
4 (k) 4
6 (k) 6
Ve↵ (k) =
H +
=
H
+
H
+
H + ··· ,
2n
4
2
2
k
2
2
4
8k
n=2
(2.1)
• RG flow from UV to IR:
Vint êk 4
Vk êk 4
Vk êk 4
VUV êk 4
VIR êk 4
–3–
VUV êk 4
VIR êk 4
Vint êk 4
Hêk
Hêk
• unique minimum between Λ and kEW
• smaller λ6(Λ)
• EW minimum forms in the IR @ 246GeV
• ‘meta-stable’ behavior for intermediate k
rting from a stable potential in the UV di↵erent flows to the IR
EW minimum forms in the IR @ 246GeV
•
minimum are possible. In the left panel we show a situation wher
during the whole
flow between
from stability
k=⇤
topotential
the point
where the electrow
distinguish
of UV
and IR potential!
rresponds to region I in Fig. 6. For smaller initial values of
co
turns positive first, then the scale dependent potential is always stable during the
Stability
with Higher-Dimensional
RG evolution
in our approximation,
cf. left panel in Fig. 3.Operators
On the other hand, if
V UV
V eff
H
H
2
Figure
2.
Left:
Sketch
of
the
possible
shape
of
V
with
µ
(⇤) > shapes
0 but 4 (⇤) < 0. Potentials
UV
• UV potential and effective potential can have quite different
are stabilized by 6 (⇤) smaller (red) and larger (blue) than 24 (⇤)⇤2 /(3µ2 (⇤)), respectively.
Right: Sketch of di↵erent possible shapes of the e↵ective potential Ve↵ (k ⇡ 0) that may occur
presence of higher-dimensional operators modifies UV potential in general way
•depending
on the size of 6 (⇤), all corresponding to a stable UV-potential with a single minimum
at H ⇡ 0 as the blue curve in the left panel. Note that the resolution of this sketch is not high
higher-dimensional
contribute
to RG
flow from
of renormalizable
operators two
•enough
to display the operators
o↵set of the
electroweak
vacuum
H = 0, the corresponding
minima appear as one at H = 0.
• take this into account with scale-dependent effective potential Veff(k;H):
both the scale k as well as the field value, such that
V (k = ⇤; H) = VUV
•
and
V (k = 0; H) = Ve↵ .
(2.15)
– 10 –of the e↵ective potential can be followed
In this manner, the (meta-)stability properties
follow (meta-)stability properties in scale-dependent manner
in a scale-dependent manner.
➡
Finally,
let us note another
subtlety
in the comparison
between the perturbative
need
non-perturbative
approach
→ functional
RG
and non-perturbative approaches. The perturbative approach usually relies on a mass-
Gauged Higgs-Top Model
already observe a similar perturbative flow toward negative 4 at high scales [9]. The running of the top Yukaw
coupling to smaller values in the ultraviolet is included when we add an SU (3) gauge sector. Correspondingly, w
Gauged Higgs-Top Model - 1-Loop Running
investigate the Euclidean action defined at the UV-cuto↵ scale ⇤
2
3
nf
Z
ny
X
X
1
1
y
2
/ j + ip
5
S⇤ = d4 x 4 Fµ⌫ F µ⌫ + (@µ ') + Ve↵ (⇤) + i
' j j5 ,
(
jD
4
2
2 j=1
j=1
1.0
SM couplings
running couplings
y
with an SU1.0
(Nc ) gauge field (Nc = 3), a real scalar ', Dirac fermions
the covariant
derivative D
j for nf flavors,
Higgs-top
model
g3
yt
including the
Ve↵ . We assume that ny < nf of the fermi
0.8
0.8 strong coupling gs , and the e↵ective scalar potential
species are heavy and shall have a degenerate large Yukawa coupling y. The remaining nf ny flavors ha
g2
0.6
negligible Yukawa
couplings.
The e↵ective scalar potential 0.6
is now expanded in terms of dimensionless couplin
g1
2n as
0.4
0.4
✓ 2 ◆n
2
X
0.2
µ(k) 2
'
2n (k)
0.2
V
(k)
=
'
+
.
(
e↵
m in TeV
2n
4
l
4
2
k
2 g3
l
n=20.0
0.0
102
yb
104
106
108
1010
1012
1014
1016
1018
100
5
108
10
potential shown in
1020
This potential should be compared to the Standard Model
RGE scale m in GeV
study beta functions for the
self-interaction 4 and the '6 -coupling
an (infinite)1.0series of possible higher-order couplings.
1.0
1011
1014
1017
1020
Eq.(1). As part of the potential w
6 . Here we restrict ourselves to the first
RG scale k in GeV
running couplings
running couplings
re 1: Renormalisation of the SM gauge couplings g1 = 5/3gY , g2 , g3 , of the top, bottom
g3Higgs quartic
gauged Higgs-top model
gaugedcoupling
Higgs-top
couplings (yt , yb , y ), of the
andmodel
of the Higgs mass parameter m. g3
0.8 toy model
0.8
Because
our
doesWenot
reflect
thethresholds
weak gauge
structure
Model, the gauge couplin
arameters
are defined
in the yms scheme.
include
two-loop
at the weak
scale yof the Standard
+ fiducial coupling
g1 andRG
g2 equations.
do not appear.
However,
we±1know
that ininthe
Model they have significant e↵ects: first, t
hree-loop
The thickness
indicates the
uncertainties
Mt , Standard
Mh , 3 .
0.6
0.6
g
F
gauge couplings give a significant positive contribution to 4 , balancing
the negative top Yukawa terms for sm
ckvalues
mass, we
the
followingthey
values slow
of the down
SM parameters:
0.4 and thereby a↵ect the increase of 4 toward lar
offind40.4
; second,
the growing top Yukawa
fieldg1 (M
values.
Since the variation of the weak coupling at large(56a)
energy scales is modest, we account for its e↵ec
Pl ) =0.20.6168
0.2
by gincluding
a finite
contribution in the beta functions for 4 (56b)
and
y, parametrized by a fiducial coupling gF an
2 (MPl ) = 0.5057
l
l
4
0.0
g3 (MPl ) = 0.4873 + 0.0002
3 (MZ )
0.1184
0.0007
◆
✓108
11
20
1014
1017 3 (MZ10
Mt 10
) 0.1184
yt (MPl ) = 0.3823 + 0.0051 RG scale173.35
0.0021
k in GeV
GeV
0.0007
✓
◆
Mt
100
105
4
0.0
(56c)
100
(56d)
105
108
1011
1014
RG scale k in GeV
1017
1020
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
•
momentum shells regularization. Di↵erent choices of Rk correspond to di↵erent regu2 Gauged Higgs–top model
larization schemes.
Functional
RG for Gauged Higgs-Yukawa Model
Even though
the Wetterich
equation
a simple one-loop
it is an exact
Nonperturbative
tool:vacuum
functionalstab
RG
The aim
of this paper
is has
to investigate
Higgsstructure,
mass
bounds
and
equation as there exists an exact equivalence to the full functional integral. The key
the presence
of higher-dimensional
operators
(2)and a finite ultraviolet cuto↵. In a
Use functionalingredient
RG method
asis an
tool
to obtain
for this
thatappropriate
the propagator
in the
loop ⇠ (β kfunctions:
+ Rk ) denotes1 the full (2)
1
@t k =renormalization
Tr @t Rk ( k + R
k)
to the standard perturbative running, we use functional
group
2
propagator at the scale k. From the full solution of Eq.(C.1), for instance, all correlation
R
methods
as
a
tool
to
compute
the
running
of
an
extended
set
of
operators.
all
quantum
fluctuations
in
presence
of
higher-dim
operators
⇤
RG
trajectory:
=
=
UeffLet
+ ...
‣ allows to include
k
!0
functions can be computed.
set up a toy model that allows us to study the essential features of the Standard
ain
perturbative
expansion
of thestability.
Wetterich
equation reproduces perturbation
RG
k interpolates
between
k with
‣ flowing action ΓWhile
thescale
context
of vacuum
theory to any order,
flow equation
also be
used to extract
non-perturbative
As a the
starting
point, wecan
briefly
recapitulate
the main
features of the Standard
information
[43]. Following the Wilsonian
scale
action k conmicroscopic
! introductory
⇤)idea,
: the
! dependent
S[ ]we use
k [ ]sections
at one-loop action
level. In(kthe
H for the actual Higgs
tains not just relevant or marginal couplings, but encodes e↵ects of high-scale quantum
while
' denotes
a general
full
e↵ective
action
(k !real
0) :scalar
]For
!which
[ ] can play the role of the Higgs
k [ field
fluctuations in the presence of higher-order operators.
practical calculations, this
inofour
toy model.
we arrive
our toy model
quantitatively
reprodu
infinite series
operators
in theOnce
e↵ective
action at
is truncated.
In thewhich
present
context,
‣ FRG flow equation:
Standard
Model,
we our
willapproximation
again use H of
forthe
theexact
corresponding
a useful ordering
scheme
defining
flow is givenscalar
by an Higgs field.
1
(2)
1
[ ] = action
STr{[in terms
[
]
+
R
]
(@oft Rthe
. To next-to-leading
expansion of@ttheke↵ective
of
derivatives
field.
k
k )}
k
2
2.1 Standard
Model running
Wetterich (1993)
order this gives
2
3
The
perturbative approach
starts
from
the
nf
ny usual Higgs potential, generalized to a
X
X
ZH
1
ZG a aµ⌫ 5
4 4
2
¯
¯
/
p
=
d
x
V
+
(@
H)
+
Z
i
D
+
i
ȳ
H
+
Fµ⌫ F scale
, k of all para
tive
potential
by
allowing
for
a
dependence
on
the
momentum
µ
j
j
j j
j j
‣ truncation: k
2
4
2
j=1
Including higher-dimensional
operatorsj=1
it reads
(C.2)
✓ 2 ◆n
the
wave
function
renormalizations.
For
the flow
of the e↵ectiv
2
2
X
where thecouplings
potential V and
, the
(bare)
Yukawa
couplings
ȳ
as
well
as
all
wave
function
renorj
µ(k) 2
H
µ (k) 2
2n (k)
4 (k) 4
6 (k) 6
V
(k)
=
H
+
=
H
+
H
+
HFo+
weZobtain
-functional
defining
a
-function
for
every
field
value
H.
‣ effective potential:
malizations
The
gauge-part
of
the
action
is
also
supplemented
2n
4
2
H,e↵,G areak-dependent.
2
k
2
2
4
8k
Z
e.g.,
n=2 contribution.
by a gauge-fixing
Fadeev-Popov
us writeand
down
this flowghost
of the e↵ective potential explicitly:
Inserting this ansatz into the Wetterich equation and2expanding both sides in terms
FRG
flow
of ofeffective
potential:
of this
basis
operators leads
to the -functions, i.e. the flow equations for the Yukawa
6 1
d V (H)
6
=
V (H) =
d log k 32⇡ 2 4 1 +
k4
– 31 –
⌘H
6
V 00 (H)
k 2 ZH
–3–
4
ny
X
j=1
Nc
1
1+
⌘
j
5
ȳj2 H 2
2k2 Z 2
j
3
7
7 .
5
Functional RG for Gauged Higgs-Yukawa Model
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
‣ 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)
‣ Threshold contributions allow for dynamical mass generation by SSB
pert. RG
FRG
SSB
SSB
SSB
(2)
k
+ Rk )
=
=
R
1
Ueff + . . .
Functional RG for Gauged Higgs-Yukawa Model
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
‣ 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
k !0
(2)
k
+ Rk )
=
=
R
Ueff + . . .
] ! S[ ]
]! [ ]
(@t Rk )} .
1
Wetterich (1993)
β functions for model couplings…
…(e.g. reproduce 1-loop β functions from PT, include threshold effects, higher-dim operators,…)
Gauged Higgs-Top Model - Higher-dimensional operators
VUV =
4 (⇤)
4
4
H +
6 (⇤) 6
H
2
8⇤
+ ...
• Potential at UV scale: completely stable with unique minimum at H=0
1.0
4 (⇤)
=
0.058
y
6 (⇤)
= 2.0
0.8
running couplings
running couplings
0.8
g3
1.0
0.6
0.4
0.2
0.0
100
l4
l6
108
1011
1014
RGE scale k in GeV
1017
1020
4 (⇤)
=
0.05
y
6 (⇤)
= 0.8
0.6
0.4
0.2
0.0
105
g3
100
l4
l6
105
108
1011
1014
1017
1020
RGE scale k in GeV
ure 4: Running couplings including 4 and 6 , corresponding to Eq.(7) with mh = 125 GeV. The bare pote
Potential
2ndscales.
Minimum
during(⇤)
RG=flow0.05
Potential
completely
duringand
entire
RGremain
flow bound ‣from
sfy ‣the
stability
conditionstable
of Eq.(14)
thus
belowdevelops
at all lower
Left:
4
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,
new physics would presumably have to set in beyond 1012 GeV, in order to control the strong running o
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
Higgs Mass Bounds
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
Models for High-Scale Physics
the renormalization group flow of the SM couplings shown in Eq.(7). To compute the allowed
hno determines
its
value
at
⇤.
We
obtain
such a scenario, we do not need to consider a UV-complete theory for the heavy states beyo
High-Scale
Physics
he model can come with Models
an inherent for
cuto↵,
⇤BSM
⇤. This
6 corresponds3to a hierarchy of
3 e↵ect
(⇤) heavy
+ 216scalars,
+ Nagain
(⇤b
S HS
4 (⇤)which
which the Standard Model is superseded by a model 12y
containing
will
(⇤)
=
6
•
how
to
generate
suitable
higher-dimensional
high-scale
2 (⇤) +physics?
a more fundamental model close to the Planck scale. couplings
4 (16⇡ 2from
+ 9y
45 (⇤))
4
• induce potential with λ4<0 and λ6>0
In our simple model we couple the additional scalar to the Standard Model through a Higgs po
sitive
aNSnegative
Higgs
quarticcutoff,
coupling,
which
simpleYukawa
model: introduce
heavy scalars
with inherent
e.g. @ΛBSM
=MPl is the
• top
o the e↵ective
potential
ofand
Eq.(6),
ds, each reduce the pseudo-fixed point2 value.
As
the4 top Yukawa coup
2
2
H S
S
S
2
portal: Accordingly,
14
tive contributionHiggs
increases.
a +sufficiently
and positive
Ve↵ = HS
mS
+ S large
.
2 2
2
4
⇤
-point value 6 , which is then depleted toward lower scales. In contrast,
0.010 S, the heavy scalar is stable and could constitute (a pa
Due to the reflection msymmetry
S
!
0.0006
= 10 GeV
1.0 the fluctuations
s
by
of
the
heavy
scalar.
This
implies
that
while
the
gener
0.005
l
g
matter
relic density. This
massive scalar field adds new0.0004
loop terms to the beta functi
l HL Ladditional
=0
0.8
0.000
y
l for
HL
L =>
0.56m it
is6 , bounded
from
below,
is
not
always
large
enough
to
yield
a
potentia
contributing
only
k
.
To
decouple
the
massive
modes
below
m
we
include
thresho
0.0002
S
S
0.6
-0.005
2
2 n
2 0.0000 e↵ects, the loop terms ca
orm
1/(1
+
m
/k
)
with
an
appropriate
power n.
Including µ
threshold
s. The
size
coupling
still decides whether
th
0.4
S of the quadratic Higgs
-0.010
l
n the0.2FRG scheme, as shown in App. C. If we allow for NS mass-degenerate
scalar fields with t
-0.0002
stable,
or
features
a
global
minimum
at
vanishing
Higgs
field.
-0.015
l
-0.0004 become
portal0.0interaction,
the one-loop beta functions of our toy model, Eq.(7),
l
-0.020
is nevertheless
possible
to generate
initial
corresponding
to10 a st
100
10
10
10
10
10
10
10
10 conditions
10
10
10
10
10

2 point
such
correctly.
then solve the
pseudo-fixed
equation
RG)scale
k in that
GeV1 the Higgs mass comes out
RG scale
k in GeV We 15
RGN
scale k in
GeV
4 (⇤BSM
14
S
4
2
2
4
HS
nishing
field
values.
As
an
example,
⇤ 6=+ 10
y(⇤
= its2 value
nat
+
2ny Nc y 4 we
+ 9 use
c gF +GeV, where
which
determines
⇤.
obtain
yN
c yWe
4
8⇡
4
4(1 + m2S /k 2 )3
0.25
ysical
Higgs
mass
of
125
GeV.
This
(⇤)
can
be
reached
by
0.030
6 value 3of
3

4
12y
(⇤)
+
216
(⇤)
+
N
(⇤)
m
=
10
GeV
S HS
3
4
14 GeV:
1
N
0.025
(⇤)
=
.
λ6 at
10
‣1.0 Generated
0.20
S
g
6
6
2
3
HS
l HL L = 0.255
2y+ 9y
2 (⇤)
4
(16⇡
+
45
(⇤))
table
with
the
measured
Higgs
mass
value,
we
need
=
2
+
2n
N
y
+
6n
N
108
+
90
.
0.8 potential
4
6
y
c
y
c
6
4
6
4
0.020
2
y
2
2 4
l HL L = 2.0
0.15
14
3
4
fund
BSM
l4
HS
*
6,HS
fund
BSM
l6
running couplings
S
*
6
4
6
6
10
14
18
10
12
14
16
18
10
12
14
16
18
4
14
3
4
6
HS
fund
fund
16⇡
2(1 + m /k )
l6
*
S l6,HS
0.015
0.10
A positive top Yukawa and a negative
Higgs quartic coupling, which0.010
is the case of interest for the Hig
0.4
3 coupling grows toward the infra
0.05
bounds, each reduce the pseudo-fixed
point
value.
As
the
top
Yukawa
4
6
0.005
S HS
0.2
0.00
l4
negative
contribution increases. Accordingly,
a sufficiently large and positive
value of HS is needed
l6* for a
0.000
0.0 l6
fixed-point
value 14 ⇤6 , which
is then
depleted
toward
lower
scales.
In contrast,
driven
to increasingly
-0.05
-0.005 10
4 is
6
10
100
10
10
10
1018
1010
1012
1014
1016
1018
10
1012
1014
1016
1018
values by the fluctuations of the HS
heavy scalar. This implies that while the generated value of 6 provides a p
RG scale k in GeV
RG scale k in GeV
RG scale k in GeV
0.6
l4
running couplings
S
N
& 24 ,
➡
induce stable potential with λ <0 and λ >0:
➡
for small number of new states need sizeable portal coupling
xample corresponding to
⇠ 2 and 3 additional scalars. This large valu
Summary & Outlook
•
measured Higgs mass very close to lower bound mh(Λ = MPl)
•
perturbative analysis: Higgs potential loses stability around 1010 GeV
L@GeVD
•
Ê
this statement can be relaxed:
1019
1017
Ê
¨III
Ê
1015
13
10
1011 ÊÊÊ
‣ higher-dimensional operators at UV scale Λ
II
Ê
Ê
Ê
Ê
Ê
Ê
Ê
1.5
2.0
Ê
I
Ê
9
10
0.0
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 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
✤
Question: What type of physics can predict higher-dim operators of suitable size?
‣ we have investigated simple SM extension with heavy scalars
to non-perturbative
‣ required parameter choices in simple model are at border
With this choice, the limit therefore essentially reduces to the usu
the meta-stable
region 022
as obtained
in the absence of any 6 .
JHEP 04 (2015)
and arXiv:1501.02812
Our numerical study based on a more advanced approxima
Rev.that
D 90
(2014)with
025023
and arXiv:1404.5962
App. Phys.
C shows
already
moderately
small values 6 ⇡