The vacuum structure of the
triplet-doublet model
Pedro Ferreira
ISEL and CFTC, UL
Lisbon, Portugal
PLANCK 2017, 25/05/2017
In collaboration with S. Kanemura and H. Sugiyama
THE MODEL: the Triplet-Doublet Model contains all SM fields, but
expands the scalar content, via the addition of an SU(2) triplet.
• Useful to explain both dark matter relic density and small neutrino
masses.
• The model has a richer particle spectrum than the SM, containing two
CP-even scalars h and H, a pseudoscalar A, a charged scalar H+ and a
doubly charged one, H++.
• In the Triplet-Doublet Model, vacua which spontaneously break the
electromagnetic symmetry are possible. The stability of neutral vacua is
therefore not guaranteed a priori.
• But it is possible to conclude, analytically, that under many
circunstances the stability of electroweak breaking minima is ensured by
the model itself, regardless of the values of its parameters.
THE SCALAR POTENTIAL: the model contains the usual SU(2)×U(1) doublet Φ,
but also a complex SU(2) triplet Δ .
The most general gauge invariant scalar potential with this field content is
with all parameters real. The necessary and sufficient bounded-from-below
conditions of the model are
NO POSSIBILITY OF SPONTANEOUS CP VIOLATION: all the parameters in the
potential are real - even the trilinear coupling μ, which can always be made real
through a rephasing of the Δ field.
A generic CP breaking vacuum can always be written, through adequate rephasings of
fields, as
However, the potential at such a vacuum has a trivial dependence of the complex phase
θ – there is a single term in the potential depending on θ, arising from the cubic terms.
The potential at this vacuum is found to be of the form
Minimizing with respect to θ immediately gives
no non-trivial CP-breaking phase.
sin θ = 0, and therefore there is
MODEL WITH DISCRETE Z2 SYMMETRY
Let us consider a version of the model wherein one imposes a discrete Z2 symmetry,
Δ → - Δ. This discrete symmetry eliminates the trilinear coupling between the doublet
and the triplet, and thus sets μ = 0.
The interest of this symmetric model is that it allows for good dark matter candidates,
whose lack of interactions with fermions is renormalization-protected.
Several neutral vacua are then possible (all vevs real):
N1:
W, Z, fermions massive
– massless axion
N2:
W, Z, fermions massive
– dark matter
N3:
W, Z, massive –
massless fermions
Clearly the N2 minimum is the most interesting one, providing phenomenology
compatible with observations (a massless scalar such as the one present in N1 is hard
to reconcile with experiments).
The model has no possibility of CP breaking, but there are several possible ChargeBreaking (CB) vacua:
...
In the 2HDM, neutral minima are guaranteed (at tree-level) to always be deeper than
eventual CB stationary points – what about the triplet-doublet model?
At the N2 minimum the squared scalar masses are given by
Using a bilinear formalism similar to the one developped for the 2HDM, it is possible
to find analytical formulae relating the depth of the potential at different extrema of
the potential. For the N2 case, for instance, one has:
If N2 is a minimum, all of its
squared scalar masses will be
positive and one will have
The N2 minimum is guaranteed to be completely stable (at tree-level) against
tunneling to extrema of types N1, CB1, CB2 (CB3...?). A similar conclusion is
reached for minima of type N1.
FULL MODEL WITH μ TERM
The μ-term (the scalar potential terms which are cubic in the fields) is of great
phenomenological importance. For instance, it is this term which does prevent an
axion from occurring in an N1 minimum.
But it also prevents the occurrence of an N2 minimum, and “spoils” the bilinear
formalism which has been permiting the analytical comparison between extrema.
The electroweak breaking minimum of greater interest when μ ≠ 0 is therefore of type
N1, but great care must be exercised in scanning its parameter space: the doublet-vev
and the triplet-vev have different contributions to the W and Z masses, thus
potentially spoiling agreement with electroweak precision data.
Since experimentally the value of ρ is extremely
close to 1, the triplet vev is necessarily small,
vΔ < 8 GeV
Despite the presence of the cubic term, it is possible to obtain analytical expressions for
the comparison of the value of the potential at a N1 minimum and a charge breaking
extremum.
PRELIMINARY!
The μ parameter is “hidden” in the pseudoscalar mass mA. Unlike the formulae
obtained for the μ = 0 model, these do not seem to guarantee the absolute stability of a
N1 minimum – the right-hand-side of these expressions is not guaranteed to be positive
simply because the masses mA, m+ or m++ are positive.
A preliminary numerical scan seems to indicate that:
- The N1 minimum, when it exists, is always DEEPER than any CB1 extremum.
- But, for a very small set of parameters, the same does not seem to hold for the
comparison between a N1 minimum and a CB2 extremum.
CONCLUSIONS
• The triplet-doublet model has a rich vacuum structure – despite the inability to
develop CP-breaking vacua, it can have several different types of charge breaking
ones.
• The bilinear formalism, adapted to the TDM, permits the obtention of analytical
formulae comparing the value of the potential at different extrema, as was observed in
the 2HDM (and also the singlet-doublet model).
• These formulae give, in many cases, rise to forceful conclusions: for the μ = 0 model,
for instance, they ensure the absolute stability of the N2 minimum (the minimum of
greater phenomenological interest in that model).
• Other CB extrema also need to be considered; CB3, for instance, is harder to deal
with.
• The introduction of the cubic terms in the potential greatly increases the complexity
of the calculation, and seemingly muddles the conclusions – the electroweak breaking
minimum N1 no longer seems wholly stable,for instance.