JHEP06(2016)082
where the summation runs over all types of fermions, y
f
is the Yukawa coupling to η,
N
f
is the number of flavors, t
f
is the Dynkin index of the colored fermions under SU(3)
c
(normalized to 1/2 for fundamental representation) and Q
f
is the electric charge of the
fermion. The branching ratio of η to two photons is given by
Br(η → γγ) ≈
α
P
f
y
f
N
c
N
f
Q
2
f
m
f
2
8
α
s
P
f
y
f
N
f
t
f
m
f
2
. (2.4)
For N
f
copies of an SU(2) singlet quark t
0
of charge 2/3, plus its vector-like partner
¯
t
0
,
Br(η → γγ) ≈ 0.4%. For N
f
copies of SU(2) doublets Q
0
= (t
0
, b
0
) of charge (2/3, −1/3)
and their vector-like partners, Br(η → γγ) ≈ 0.15%. These two choices will be our bench-
mark models in the following discussions. We will also assume that heavy quarks are all
degenerate in mass for simplicity.
The production cross section of η, when we add the vector-like color triplets with a
common y
f
and m
f
, is approximately
σ
pp→η
≈ 400 fb
y
f
N
eff
f
v
m
f
!
2
, (2.5)
where v = 246 GeV is the Higgs vev.
4
In models with singlet quarks t
0
i
, N
eff
f
= N
f
, while
in models with doublets Q
0
i
, N
eff
f
= 2N
f
.
In the case of t
0
or Q
0
, this can be translated into a bound on the combination of the
parameters,
m
f
=
(
y
f
N
eff
f
(245 GeV)/(1.4 → 1.9) t
0
y
f
N
eff
f
(245 GeV)/(2.2 → 3.2) (t
0
, b
0
)
, (2.6)
where the lower (higher) number in the denominator corresponds to 3 (6) fb for σ(pp → X
s
).
In cases with t
0
decaying to t + η or to t(b)+ SM gauge bosons, the new quarks have to
be heavier than ∼ 1 TeV, as we will discuss below, and so y
f
N
eff
f
& 6. To keep the theory
perturbative, (i.e. the one-loop correction to y
f
smaller than y
f
),
5
we need y
2
f
N
c
N
f
. 16π
2
.
In general one can obtain any values of κ
G
, κ
γ
desired, by including multiple particles
with different masses, hypercharges, etc., and including colorless particles. Therefore one
must view the κ
G
, κ
γ
as free parameters, only requiring that they not be so large as to
violate perturbativity in the effective theory.
Since κ
G
is induced at one loop, this type of model tends to give η a width below 1 GeV,
and does not immediately explain how a resonance could have a width of tens of GeV, as
ATLAS’s excess seems tentatively to suggest. However, the model is easily modified. For
instance, a renormalizable interaction of η with invisible particles could greatly increase
4
The prefactor of 400 fb is the cross-section for a 750 GeV Standard Model Higgs in the limit that
m
t
→ ∞. We computed this in Madgraph with the Higgs effective theory (heft) model file [10] and
assigned a K factor ∼ 2 (for 14 TeV, the K factor for ggh of a 750 GeV Higgs is around 1.8 [11]).
5
Note added. In a previous version we used a slightly stronger criterion; figure 1 now has a slightly
weaker perturbativity constraint, but of course this is not a sharp boundary in any case.
– 4 –
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