where ··· represents terms proportional to various powers of
S
, b and neglected higher-
dimensional operators. Allowing terms proportional to
S
, bm
S
leads, at dimension-6, to
only one additional operator, |H|
6
, which taken together with the ability to change the
sign of the |H|
4
operator in the SM-Higgs Lagrangian and yet maintain a stable minimum
to the Higgs potential, can have interesting implications for the order of the electroweak
phase transition [50, 59].
For the purposes of the discussion here, the |H|
4
operator can simply be absorbed
into an unobservable shift of the SM Higgs quartic-coupling: λ → λ +
1
2
a
2
. The fermionic
Higgs portal operator H
†
H ¯χχ leads to a variety of effects (see, e.g., ref. [31] for a detailed
analysis): upon EWSB, it contributes to the mass of the χ field, and allows both h¯χχ and
h
2
¯χχ couplings. The former coupling allows for χ-χ scattering via Higgs exchange, and
there is a further scattering contribution from the ( ¯χχ)
2
operator.
It is possible in this scenario to have the χ field play the role of the stable dark matter
(DM), saturating the relic density. If the mediator S is assumed to be somewhat heavier
than the χ, an EFT analysis is applicable [30, 31]: the relic abundance requirement fixes
m
S
/(aκ
S
) ∼ 0.5 TeV, while satisfying the LUX direct detection bounds [60] requires that
the χ be fairly massive, m
physical
χ
& 3 TeV. These requirements are actually in tension
with the assumptions of an EFT analysis as they demand a very large coupling aκ
S
& 2π
to avoid m
χ
> m
S
and the resulting issues with perturbative unitarity [61–63] for the
EFT description of the non-relativistic DM freeze-out. This essentially means that, with
the finely-tuned exception of m
χ
∼ m
h
/2 [30, 31], the heavy-mediator scenario is ruled
out here. On the other hand, the scenario with a light mediator, m
S
< m
χ
, is still
viable as the direct detection constraints can be avoided while maintaining the correct relic
abundance [30]. Of course, if we were to abandon the DM interpretation for χ, and allow it
to decay sufficiently quickly via some suitable modification to this model, the couplings a
and/or κ
S
can be freely dialed down without overclosing the universe, thereby alleviating
the direct detection constraints and allowing light m
χ
.
In the EFT description for this model, the operator
1
2
(∂
µ
|H|
2
)
2
leads to a “Higgs
oblique” correction to the Higgstrahlung process; i.e., the operator induces a Higgs wave-
function renormalization δZ
h
= a
2
v
2
/m
2
S
, which modifies the cross-section by ∆σ
Zh
/σ
Zh
=
δZ
h
[26, 49]. With a sensitivity of ∆σ
Zh
/σ
Zh
∼ 0.5% (corresponding closely to the CEPC
and FCC-ee sensitivities in table 1), values of m
S
/a . 2.5 TeV could be ruled out at 95%
confidence, with 5-σ discovery reach up to m
S
/a ∼ 1.6 TeV [49]. (The limits on m
S
/a just
quoted are slightly weaker than those in figure 1 of ref. [50], wherein a combined fit to all
projected ILC Higgs coupling measurements is considered.)
In this model, the electroweak precision observables S and T are generated by one-
loop running of the operators between µ = m
S
and µ = m
W
, leading to a mixing of the
Higgs oblique operator with the operators responsible for giving the S and T parameters.
This one-loop running has been computed in ref. [50], which considers the UV completion
above in the limit that κ
S
→ 0 (and with no assumption on the sizes of
S
, b). The
leading contributions to the S and T parameters are found to be log-enhanced, but linearly
proportional to the high-scale (µ = m
S
) value of the Wilson coefficient of the Higgs oblique
operator, and suppressed by a loop factor.
– 6 –