JHEP04(2017)155
Published for SISSA by Springer
Received: February 16, 2017
Revised: March 23, 2017
Accepted: April 18, 2017
Published: April 27, 2017
Constraints on the trilinear Higgs self coupling from
precision observables
G. Degrassi,
a
M. Fedele
b
and P.P. Giardino
c
a
Dipartimento di Matematica e Fisica, Universit`a di Roma Tre and
INFN — Sezione di Roma Tre,
I-00146 Rome, Italy
b
Dipartimento di Fisica, Universit`a di Roma “La Sapienza” and
INFN — Sezione di Roma,
I-00185 Rome, Italy
c
Physics Department, Brookhaven National Laboratory,
Upton, New York 11973, U.S.A.
E-mail: degrassi@fis.uniroma3.it, marco.fedele@uniroma1.it,
pgiardino@bnl.gov
Abstract: We present the constraints on the trilinear Higgs self coupling that arise from
loop effects in the W boson mass and the effective sine predictions. We compute the
contributions to these precision observables of two-loop diagrams featuring an anomalous
trilinear Higgs self coupling. We explicitly show that the same anomalous contributions
are found if the analysis of m
W
and sin
2
θ
lep
eff
is performed in a theory in which the scalar
potential in the Standard Model Lagrangian is modified by an (in)finite tower of (Φ
†
Φ)
n
terms with Φ the Higgs doublet. We find that the bounds on the trilinear Higgs self coupling
from precision observables are competitive with those coming from Higgs pair production.
Keywords: Beyond Standard Model, Higgs Physics
ArXiv ePrint: 1702.01737
Open Access,
c
The Authors.
Article funded by SCOAP
3
.
doi:10.1007/JHEP04(2017)155
JHEP04(2017)155
Contents
1 Introduction 1
2 λ
3
-dependent contributions in m
W
and sin
2
θ
lep
eff
3
3 Equivalence with a (Φ
†
Φ)
n
theory 5
4 Results 8
5 Conclusions 10
A Anomalous contributions in ∆ˆr
(2)
W
and Y
(2)
M S
11
1 Introduction
The discovery of a new scalar resonance with a mass around 125 GeV at the Large Hadron
Collider (LHC) [1, 2] opened a new era in high-energy particle physics. The study of the
properties of this particle provides strong evidence that it is the Higgs boson of the Standard
Model (SM), i.e., a scalar CP-even state whose coupling to the other known particles has
a SM-like structure and a strength proportional to their masses [3–5]. At present, the
combined analysis based on 7 and 8 TeV LHC data sets [5] shows that the couplings with
the vector bosons are found to be compatible with those expected from the SM within a
∼ 10% uncertainty, while in the case of the heaviest SM fermions (the top, the bottom
quarks and the τ lepton) the compatibility is achieved with an uncertainty of ∼ 15 −20%.
Concerning the future, the best present estimates [6, 7] indicate that at the end of the LHC
Run-2 at
√
s = 13 − 14 TeV center-of-mass-energy, the fit of the Higgs boson couplings to
the vector bosons is expected to reach a ∼ 5% precision with 300 fb
−1
luminosity, while
the corresponding ones for the fermions, with the exception of the µ lepton, can reach
∼ 10 −15% precision. Similar estimates for the end of the High Luminosity option indicate
a reduction of these numbers by a factor ∼ 2.
The study of the Higgs self interactions, coming from the scalar potential part in the
Standard Model (SM) Lagrangian, is in a completely different status. In the SM, the Higgs
potential in the unitary gauge reads
V (φ
1
) =
m
2
H
2
φ
2
1
+ λ
3
vφ
3
1
+
λ
4
4
φ
4
1
(1.1)
where the Higgs mass (m
H
) and the trilinear (λ
3
) and quartic (λ
4
) interactions are linked
by the relations λ
SM
4
= λ
SM
3
= λ = m
2
H
/(2 v
2
), where v = (
√
2 G
µ
)
−1/2
is the vacuum
expectation value, and λ is the coefficient of the (Φ
†
Φ)
2
interaction, Φ being the Higgs
doublet field.
– 1 –
JHEP04(2017)155
The experimental verification of these relations, that fully characterize the SM as a
renormalizable Quantum Field Theory, relies on the measurements of processes featuring at
least two Higgs bosons in the final state. However, since the cross sections for this kind of
processes are quite small, constraining the Higgs self interaction couplings within few times
their predicted SM value is already extremely challenging. In particular, information on λ
3
can be obtained from Higgs pair production with the present bounds on this reaction from
8 TeV data that allow to constrain λ
3
within O(±(15 − 20)λ
SM
3
) [8–11]. At
√
s = 13 TeV,
the Higgs pair production cross section, in the SM, is around 35 fb in the gluon-fusion
channel [12–21] and even smaller in other production mechanisms [22, 23] that suggests,
assuming an integrated luminosity of 3000 fb
−1
, that it will be possible to exclude at the
LHC only values in the range λ
3
< −1.3 λ
SM
3
and λ
3
> 8.7 λ
SM
3
via the b
¯
bγγ signatures [24]
or λ
3
< −4 λ
SM
3
and λ
3
> 12 λ
SM
3
including also b
¯
bτ ¯τ signatures [25]. Concerning the
quartic Higgs self-coupling λ
4
, its measurement via triple Higgs production seems beyond
the reach of the LHC [26, 27] due to the smallness of the corresponding cross section
(around 0.1 fb) [20].
In order to constrain the trilinear Higgs self coupling, a complementary strategy based
on the precise measurements of single Higgs production and decay processes was recently
proposed. In this approach the effects induced at the loop level on single Higgs processes
by a modified λ
3
coupling are studied. This approach builds on the assumption that New
Physics (NP) couples to the SM via the Higgs potential in such a way that the lowest-
order Higgs couplings to the other fields of the SM (and in particular to the top quark
and vector bosons) are still given by the SM prescriptions or, equivalently, modifications
to these couplings are so small that do not swamp the loop effects one is considering. This
strategy was first applied to ZH production at an e
+
e
−
collider in ref. [28] and later to
Higgs production and decay modes at the LHC [29–31].
The aim of this work is twofold. On the one side we apply the same strategy to the
precise measurements of the W boson mass, m
W
, and the effective sine, sin
2
θ
lep
eff
. In order
to constrain λ
3
we look for effects induced by an anomalous Higgs trilinear coupling at the
loop level in the predictions of m
W
and sin
2
θ
lep
eff
. Following the approach of ref. [29] we
parametrize the effect of NP at the weak scale via a single parameter κ
λ
, i.e. the rescaling
of the SM trilinear coupling λ
SM
3
, so that the φ
3
1
interaction in the potential is given by
V
φ
3
1
= λ
3
v φ
3
1
≡ κ
λ
λ
SM
3
v φ
3
1
, λ
SM
3
≡
G
µ
√
2
m
2
H
, (1.2)
and compute, in the unitary gauge, the effects induced by κ
λ
in the two-loop W and Z
boson self-energies, which are the relevant quantities entering in the two-loop determination
of m
W
and sin
2
θ
lep
eff
. On the other side we specify better the anomalous coupling approach
employed above by showing that, at the order we are working, i.e. at the two-loop level, it
is equivalent to the use of a SM Lagrangian with a scalar potential given by an (in)finite
tower of (Φ
†
Φ)
n
terms. Furthermore, we show that the use of the unitary gauge in the
anomalous coupling approach does not introduce any gauge-dependent problematics.
The paper is organised as follows. In section 2 we discuss the contributions induced by
an anomalous Higgs trilinear coupling in m
W
and sin
2
θ
lep
eff
. Section 3 is devoted to show that
– 2 –
JHEP04(2017)155
the addition to the SM Lagrangian of (Φ
†
Φ)
n
terms gives rise to the same contributions.
In the following section we discuss the constraints on λ
3
that can be obtained from the
current data. In the last section we summarise and draw our conclusions.
2 λ
3
-dependent contributions in m
W
and sin
2
θ
lep
eff
We consider a Beyond-the-Standard-Model (BSM) scenario, described at low energy by
the SM Lagrangian with a modified scalar potential. We further assume that only Higgs
self couplings will be affected by this modified potential while the strength of the couplings
of the Higgs to fermions and vector bosons will not change with respect to its SM value,
or, equivalently, that any modification of these couplings is going to induce effects much
smaller than the ones coming from the “deformation” of the Higgs self couplings.
In the M S formulation of the radiative corrections [32–34] the theoretical predictions of
m
W
and sin
2
θ
lep
eff
are expressed in terms of the pole mass of the particles, the MS Weinberg
angle
ˆ
θ
W
(µ) and the M S electromagnetic coupling ˆα(µ), defined at the ’t-Hooft mass scale
µ, usually chosen to be equal to m
Z
. In particular, given the radiative parameters ∆ˆr
W
,
∆ˆα, Y
MS
defined through (sin
2
ˆ
θ
W
(m
Z
) ≡ ˆs
2
) [35]
G
µ
√
2
=
π ˆα(m
Z
)
2m
2
W
ˆs
2
(1 + ∆ˆr
W
) , ˆα(m
Z
) =
α
1 − ∆ˆα(m
Z
)
,
ˆρ ≡
m
2
W
m
2
Z
ˆc
2
=
1
1 − Y
MS
, (2.1)
with ˆc
2
= 1 − ˆs
2
, m
W
is obtained from m
Z
, α, G
µ
via
m
2
W
=
ˆρ m
2
Z
2
1 +
"
1 −
4
ˆ
A
2
m
2
Z
ˆρ
(1 + ∆ˆr
W
)
#
1/2
, (2.2)
where
ˆ
A = (π ˆα(m
Z
)/(
√
2G
µ
))
1/2
, while the effective sine is related to ˆs
2
via
sin
2
θ
lep
eff
=
ˆ
k
`
(m
2
Z
)ˆs
2
,
ˆ
k
`
(m
2
Z
) = 1 + δ
ˆ
k
`
(m
2
Z
), (2.3)
where
ˆ
k
`
(q
2
) is an electroweak form factor
1
(see ref. [36]) and
ˆs
2
=
1
2
1 −
"
1 −
4
ˆ
A
2
m
2
Z
ˆρ
(1 + ∆ˆr
W
)
#
1/2
. (2.4)
In our BSM scenario the modifications of the scalar potential affect the radiative
parameters ∆ˆr
W
and Y
MS
at the two-loop level while ∆ˆα and δ
ˆ
k
`
(m
2
Z
) are going to be
affected only at three loops. Recalling that the present knowledge of m
W
and sin
2
θ
lep
eff
in the SM includes the complete two-loop corrections, we are going to discuss only the
1
In our M S formulation the top contribution is not decoupled. Then
ˆ
k is very close to 1 and sin
2
θ
lep
eff
can be safely identified with ˆs
2
[36].
– 3 –
JHEP04(2017)155
φ
1
φ
1
W W W W W W
φ
1
φ
1
W W
φ
1
+
φ
1
φ
1
φ
1
φ
1
φ
1
φ
1
=
φ
1
a) b) c) d)
φ
1
φ
1
φ
1
φ
1
φ
1
φ
1
φ
1
e
1
) e
2
)e)
W W W
Figure 1. Two-loop λ
3
-and-λ
4
-dependent diagrams in the W self-energy, in the unitary gauge.
The dark blob represent the insertion of the modified diagrams in the one-loop Higgs self energy,
shown in the second row. The black point represents either an anomalous λ
3
or λ
4
.
modifications induced in ∆ˆr
W
and Y
MS
. The two-loop contribution to ∆ˆr
W
and Y
MS
can
be expressed as [35]
∆ˆr
(2)
W
=
Re A
(2)
W W
(m
2
W
)
m
2
W
−
A
(2)
W W
(0)
m
2
W
+ . . . (2.5)
Y
(2)
MS
= Re
"
A
(2)
W W
(m
2
W
)
m
2
W
−
A
(2)
ZZ
(m
2
Z
)
m
2
Z
#
+ . . . (2.6)
where A
W W
(A
ZZ
) is the term proportional to the metric tensor in the W (Z) self energy
with the superscript indicating the loop order, and the dots represent additional two-loop
contributions that are not sensitive to a modification of the scalar potential.
From the knowledge of the additional contributions induced in ∆ˆr
(2)
W
and Y
(2)
MS
one can
easily obtain the modification of the radiative parameters ∆r and κ
e
(m
2
Z
) of the On-Shell
(OS) scheme [37]. Considering only new contributions from the modified scalar potential
one can write
∆r
(2)
= ∆ˆr
(2)
W
−
c
2
s
2
Y
(2)
MS
, (2.7)
where c
2
≡ m
2
W
/m
2
Z
, s
2
= 1 − c
2
with ∆r being the radiative parameter entering the
m
W
−m
Z
interdependence. The effective sine is related to s
2
in the OS scheme via sin
2
θ
lep
eff
=
κ
e
(m
2
Z
)s
2
and for the new contributions in κ
e
(m
2
Z
) one can write
κ
(2)
e
(m
2
Z
) = 1 −
c
2
s
2
Y
(2)
MS
. (2.8)
The new contribution in the self energies in eqs. (2.5), (2.6) can be parametrized just
by a modification of the trilinear coupling as described in eq. (1.2). In order to correctly
identify the effects related to the φ
3
1
interaction we follow ref. [29] and work in the unitary
gauge. Here we discuss the W self energy but an identical analysis can be done also for
the Z self energy.
– 4 –