5.1. Leading Search Modes at the LHC
Shortly after our EWBG [
33] and alignment [
30] studies, we capitalized on the
and
couplings and proposed [
62] the associated production mechanism, that is, the two diagrams on the left side of
Figure 6,
where production is due to
, while
decay can go through
or
, giving rise to
(Same-Sign Top plus
c-jet), or
(Triple-Top) signatures. The discovery potential for
already looks promising [
62] with LHC Run 2 data, while the more exquisite Triple-Top, at higher threshold and with tiny SM cross section, can be explored at the High-Luminosity LHC (HL-LHC). See Ref. [
63] for further discussion. Note that
has a SM cross section at
fb, which is about an order of magnitude larger than triple-top in SM, and has been fervently searched for by both ATLAS and CMS, which puts constraints [
64] on
and
.
The down-type Yukawa interaction is analogous to Equation (
4), while the corresponding charged Higgs Yukawa coupling can be found in Ref. [
63]. It is curious that it took some while for us to come up with [
65] the novel
associated production process illustrated in the two diagrams on the right side of
Figure 6, that is,
where
enters the
coupling, while
goes through
. It is not surprising that the latter coupling has an associated
CKM factor, but a bit counter-intuitively (compared with 2HDM II), the former also has the
CKM factor rather than
, hence it is enhanced by
at amplitude level [
65] . Furthermore, association with the
b quark means the threshold is lower than
production, hence it is generally more efficient.
Our study [
65] did not find severe backgrounds for the
signature, but we await experimental scrutiny to find out whether there are some yet unspecified background. With
and
largely unexplored, and keeping in mind each one of them could possibly drive EWBG, we urge the ATLAS and CMS experiments to make a serious effort to search for the processes of Equations (
27) and (
28). We note that, even if the signatures are discovered, reconstructing the
H,
A and
bosons would be the next challenge, let alone disentangling the CPV phase of
down the line.
We remark that in our collider studies, we always use the 2HDMC package [
66,
67] to check parameter space, that many standard criteria such as positivity, perturbativity and unitarity, as well as electroweak precision observables, are satisfied.
5.2. Glimpse of the Coming New Flavor Era
What hides H, A, effects so well from our view?
In this subsection we will show that the conservative pattern of Equation (
26), which respects the mass-mixing hierarchy of SM Yukawa couplings as revealed by the
eEDM cancellation mechanism,
does hide exotic Higgs effects rather well in the flavor sector.
After pointing out [
54] the importance of
, we utilized the two-loop mechanism, analogous to
Figure 3 and
Figure 4, to explore
[
68]; a sizable
, together with
, could make it dominate over one-loop. The same mechanism was later applied to
.
With myriads of extra Yukawa couplings for up- and down-type quarks and charged leptons, we recently made a survey [
60] of processes of interest, starting with
and
. We saturate Equation (
26) and take:
,
,
,
, except
, as we take the relatively low
GeV. We see from the left side entries of
Figure 7 that both
and
could be discovered in the near future, by MEG II and Belle II, respectively. The
and
decays would be dominated by dipole transition, with
falling outside of Belle II sensitivity. Particularly interesting [
60] may be
conversion, where COMET at KEK and Mu2e at Fermilab aim for up to 6 orders of magnitude improvement. If realized, these experiments have the potential to disentangle various extra Yukawa couplings by utilizing different nuclei.
Turning to
B decays with leptons in the final state, we contrast g2HDM with five spectacular projections [
69] from leptoquarks (LQ) of the PS
model (three copies of Pati-Salam symmetry) motivated by the “B anomalies” (for a description and critique, see Ref. [
21]), which are illustrated by grey bands on the upper side of
Figure 7. The B anomalies are large effects, hence lead to spectacular projections, including
that falls into the Belle II range. We comment on the correlated modes of
and
below, where the studies are a bit more difficult. However, the two other modes,
(note that it was LHCb measurement that pushed down the PS
projection.) and
are very interesting. For the latter, BaBar has shown the way with full hadronic tag of the other
B, and one can just count events in the
window, while LHCb has demonstrated they can do something similar, that is, with full kinematic control from a decaying excited
parent. Surprisingly, Belle has not shown anything so far, and
would be a competition between LHCb and Belle II in the future.
We note from
Figure 7 that
and
have SM projections that are orders of magnitude below experimental sensitivity, so PS
enhancement is certainly motivating. However, for the “middle-ground” modes in
Figure 7, g2HDM projections that are illustrated by the red downward arrow are even further away from experimental scrutiny. This illustrates the efficacy of Equation (
26) in hiding the exotic Higgs boson effects in g2HDM, making
and
even harder to see, although they certainly should be searched for.
Finally, we come to the last four modes on the far right of
Figure 7:
and
. The former two have been vigorously searched for by LHCb and CMS, with LHCb holding the upper hand so far, and with indication that
is slightly below SM expectation, while
is not yet measured.
6 The latter two modes have been searched for at the B factories, with
providing [
70] one of the two important bounds on
in 2HDM II, where the current result is consistent with SM expectation. The
mode has been under Belle scrutiny lately [
13], and will be a mode of great interest at Belle II, especially in g2HDM. It was in fact the study [
71] of this decay that clarified for us some intricacies of
effects in g2HDM that differs from 2HDM II. However, it still took us some time to propose [
65] the
process of Equation (
28), which enjoys CKM enhancement.
We had pointed out earlier [
72] that g2HDM could in principle make the ratio
deviate from the SM expectation of 0.0045, where 2HDM II shares the same value [
70]. However, it was only by checking explicitly [
71] that we found that, indeed,
would be SM-like, but
could be more easily shifted, even when one respects Equation (
26). Besides a large CKM enhancement on the quark side (
transition), this is in part due to some intricacy that the
neutrino flavor is not measured in this decay, hence allowing
to enter. It would be exciting if Belle II finds
, as it would not only be BSM, but would rule out 2HDM II as well.
These last four modes share the virtue that they have SM expectations that have been driving experimental measurement. Thus, BSM effects are more effectively probed through interference, and there is much to look forward to in the near future.
Let us comment on whether g2HDM itself could impact on the B-anomalies in general. Early on, g2HDM (called 2HDM III) was proposed [
73] as a possible solution to the “BaBar anomaly”. However, as stated, the B-anomalies are large, tree-level effects. It was subsequently pointed out [
74] that this approach ran into an issue with the
lifetime by a constraint on
, hence was disfavored. The issue has recently been reopened by the loosening [
75,
76] of the
constraint, and it remains to be seen [
77,
78,
79] whether g2HDM can handle
,
anomalies. However, in general, the conservative Equation (
26), which is consistent with fermion mass-mixing hierarchies as well as
eEDM constraint, cannot easily account for large tree-level effects.
5.3. Lattice Connection: Phase Transition and Landau Ghost
We have mentioned that the first order electroweak phase transition demands Higgs quartic couplings of the g2HDM Higgs potential. At the same time, also ought to be , otherwise a large would damp away dynamical effects such as EWPT, as well as EWBG itself. Thus, GeV sets the electroweak scale, and all other dimensionless parameters of the Higgs potential are , which is why the exotic Higgs bosons populate 300 to 600 GeV, ripe for the LHC to explore.
The
Higgs quartics are not weak, bringing in two aspects to ponder. The first is to go beyond one-loop resumed effective potential [
29] and put the Higgs potential on the lattice [
80], to check nonperturbatively how the first order EWPT occurs. Though there are quite a few Higgs quartic couplings in g2HDM, this is a question of interest in its own right, and it is quite timely to check the Higgs quartic coupling parameter space that support first order EWPT.
A second issue is even more dynamical: the Landau pole of these
Higgs quartics. A simple estimate in Ref. [
30] gave 10–20 TeV, which is rather interesting. It implies some strong interaction and one would have to reconsider the theoretical framework—another issue to be studied on the lattice. Establishing the strengthening of the quartic couplings with scale would imply New Physics at some higher scale beyond, which could justify the 100 TeV pp collider. Although it would be a challenge to theorists to find a theory that can accommodate this, one might bring back SUSY and reconsider it above this scale.
With enough experimental progress, one could in principle interface the lattice study with experimental development on the exotic Higgs search, for example, exploring exotic Higgs scattering processes to actually measure the increase in quartic coupling strength.
In any case, searching for sub-TeV exotic Higgs bosons at the LHC is mandatory.
5.4. Possible Implications of Muon G-2 in g2HDM
After much effort, the Fermilab Muon g-2 experiment recently confirmed [
81] the previous measurement at BNL, and the combined result deviates from the “consensus” theory prediction [
82] by 4.2
. Thus, the muon
anomaly has to be taken seriously. In fact, g2HDM can quite easily handle it,
if one is willing to deviate from the conservative
estimate of Equation (
26). As illustrated in
Figure 8, the one-loop mechanism readily handles the observed discrepancy with, for example,
GeV (with
heavier) and
, which is 20 times
. The one-loop mechanism suffers chiral suppression, which is one of the reasons why the two loop mechanism could win over. However, having
in the loop helps, as it is not so severely chiral-suppressed.
A large
or
would enter
via the two-loop mechanism with the help of
, analogous to our previous discussion. However, what may be surprising [
83] is that, through the production chain [
84] of
the recent search bounds by CMS [
85] turn out to be more stringent on the product
than from the recent Belle measurement of
[
86]. This means that a hint could well appear with full Run 2 data, as CMS only used 36 fb
data at 13 TeV in Ref. [
85].
With
and
as before, the present bound on
is about 0.1, which is still efficient for EWBG. A finite
value would have
in the final state and dilute the
branching fraction, hence allow
to be larger (although
constraint would kick in). With
and
both sizable, one could have [
83] the novel signatures of
,
via
weak decay, showing the potential implications of the muon
anomaly.
We found [
87] further profound impact of the muon
anomaly: a possible revival of muon physics. If one replaces the final
in
Figure 8 by
e, the electron, one has a one-loop mechanism for
, which would be handily suppressed by Equation (
26), but now can allow MEG II to probe
down to
strength. If MEG II makes a discovery, it can be followed up by COMET/Mu2e for
, which now can even probe
by using various different nuclei. For
physics, Belle II can readily probe down to
with
, while
can now probe down to
, which seems quite exciting.
We had already been investigating the EDM of muon and tau, assuming Equation (
26). It is easy to note that the same one-loop diagram of
Figure 8 would give rise to
with complex
. We note in passing that, while this does not help
EDM, we find that [
61]
EDM can be enhanced to
cm, within the range of a proposed experiment at PSI, which adds to the “renaissance of muon physics”.
We stress, however, that this one-loop muon
mechanism, though exciting (and not impossible), would actually make
Nature appear rather “whimsical” [
61]. Judging from the pattern of hiding exotic scalar effects so well through the fermion mass-mixing hierarchies, we think that Equation (
26) is more likely to be realized.