Current Status of Top-Specific Variant Axion Model

The invisible variant axion model is very attractive as it is free from the domain wall problem. This model requires at least two Higgs doublets at the electroweak scale, with one Higgs doublet carrying a nonzero Peccei-Quinn (PQ) charge and the other being neutral under the PQ $\text U(1)$ symmetry. We consider a scenario where only the right-handed top quark is charged under the PQ symmetry and couples with the PQ-charged Higgs doublet. As a general prediction of this model, the top quark can decay to the observed standard model-like Higgs boson $h$ and the charm or up quark, $t\to h~ c/u$, which recently exhibit slight excesses at LHC Run-I and Run-II. It will soon be testable at the LHC Run-II. If the rare top decay excess stays at the observed central value, we show that $\tan \beta \sim 1$ or smaller is preferred by the Higgs data. The chiral nature of the Higgs flavor-changing interaction is a distinctive feature, and can be tested using the angular distribution of the $t \to ch$ decays at the LHC.


I. INTRODUCTION
The strong CP problem can be elegantly solved by the Peccei-Quinn (PQ) mechanism [1], where a U(1) P Q symmetry is employed to rotate away θ QCD , the CP-violating phase in QCD.
Not manifest in the standard model (SM), the PQ symmetry must be broken spontaneously, thereby predicting the existence of a Nambu-Goldstone boson. Since the PQ symmetry is anomalous, the additional light degree of freedom associated with the symmetry breaking is a massive pseudo Nambu-Goldstone boson, the axion [2,3]. Dynamics of the axion is characterized by the axion decay constant f a . The lower bound on f a is obtained from axion helioscopes and astronomical observations to be f a > ∼ 10 9 GeV (see, for example, Ref. [4]).
Moreover, coherent oscillations of the axion field can play the role of cold dark matter in the present Universe [5][6][7], from which one determines f a ∼ 10 10−12 GeV [8,9]. if the axion is the dominant component of dark matter. This beautiful mechanism, however, suffers from the problem of domain wall formation in the early Universe. This is because the model has N DW = 3 discrete vacua related to the number of fermion families.
The variant axion model introduced in Refs. [10,11] is an interesting axion model as it is free from the above-mentioned domain wall problem. This is achieved by allowing only one right-handed quark to carry a PQ charge and thus rendering a unique vacuum (N DW = 1) [12]. For consistency, the model requires two Higgs doublet fields, one of which is also charged under the PQ symmetry. As a result, there is a non-trivial flavor structure in the Yukawa couplings [13] that can lead to flavor-changing neutral-current (FCNC) couplings of the Higgs bosons to at least quarks. Besides, such FCNC couplings depend on the chirality of fermions not seen in the common two-Higgs doublet models (2HDM's). Therefore, the variant axion model exhibits interesting and distinctive phenomena in flavor physics at low energies.
In Refs. [13,14], we had considered such a 2HDM with the PQ symmetry and assign a nonzero charge to the right-handed top quark, thus dubbed the top-specific variant axion model. Previously, we had performed the parameter scan based on the constraints on the mixing parameters in the Higgs sector using the LHC Run-I data [14]. In particular, there was an interesting excess in the h → τ µ decay [15,16] at that time. Therefore, we had focused on the compatibility of this model to the excess and discussed the connection between the quark sector and the lepton sector. Recently, the new Run-II results negate the excess and have a result being well consistent with the SM prediction [17]. In this letter, we aim to update the parameter fitting using the current Higgs data, and study its primary signature of the t → h c/u decays that are being probed at the LHC.
Recently, the ATLAS Collaboration has reported a slight excess in the t → ch channel at 1.5σ level, BR(t → ch) = (0.22 ± 0.14)%, using the full set of the LHC Run-I data [18].
That excess is consistent with the corresponding CMS observation at LHC Run-I, given by BR(t → ch) < 0.40% (or BR(t → uh) < 0.55%) at 95% confidence level (C.L.) [19]. More recently, an ATLAS analysis based on the LHC Run-II data is published and also suggests an interesting excess in the t → ch decay [20], where the observed results set an upper bound We also show how the chirality nature in the FCNC's can be probed by studying the angular distribution of the t → ch decay at LHC Run-II.
A crude estimate for the future sensitivity on the t → ch decay without optimization was given as 2 × 10 −3 (5 × 10 −4 ) in the lepton channels and 5 × 10 −4 (2 × 10 −4 ) in the photon channels [21-23], assuming an integrated luminosity of 300 fb −1 (3 ab −1 ) in Run-III. In fact, one notes that the current sensitivity already approaches a similar level by combining the γγ, multi-lepton, and bb modes, with just the LHC Run-I (7 and 8-TeV) data of O(25)fb −1 .
We can expect the final combined sensitivity at 3 ab −1 to reach below 0.01%. This paper is organized as follows. Section II discusses the structure of the Higgs sector along with the FCNC couplings of the SM-like Higgs boson to fermions in the top-specific variant axion model. We also discuss the expected sizes of FCNC decays and theoretical constraints on this model. In Section III, we perform a χ 2 -fit analysis based on the latest Higgs signal strength data at the LHC. In Section IV, we discuss how to confirm this model through the angular distribution of t → c h decay and show the expected sensitivity corresponding to the scenario considered in this work. Conclusions are given in Section V.

II. TOP-SPECIFIC VARIANT AXION MODEL
As a minimal setup of the variant axion model, we introduce two Higgs doublet fields Φ 1 and Φ 2 and a scalar field σ with PQ charges 0, −1 and 1, respectively. The gauge singlet scalar σ gets a vacuum expectation value (VEV) f a and breaks the PQ symmetry spontaneously at a high energy scale. It therefore does not play much a role at low energies.
In the quark sector, we assume that only the right-handed top quark field t R possesses a nonzero PQ charge of −1. Note that we can additionally assign nonzero PQ charges to leptons as well, as they do not contribute to the number of domain walls N DW [12]. In this paper, we focus on the scenario where the leptons have no PQ charges. The interesting phenomenology of the case when the leptons also carry PQ charges can be found in our previous analysis [14].
The most general renormalizable Higgs potential obeying the PQ symmetry with the above PQ charge assignments is, as already given in Refs. [13,14,24]: where the σ field has been integrated out. The m 2 12 terms softly violate the PQ symmetry (as can be derived from a UV-complete Lagrangian [13]), and can be made real and positive through a rotation of the PQ symmetry. All the other terms respect the PQ symmetry and their associated parameters (m 2 11 , m 2 22 , and λ 1,2,3,4 ) are real. After the electroweak symmetry breaking, each Φ i acquires a respective VEV v i and can be written in terms of component fields as Φ (246 GeV) 2 . We now rotate the Higgs doublets to the so-called Higgs basis [25], where only one of the doublets (Φ SM ) has a nonzero VEV: where G ± and G 0 are the would-be Nambu-Goldstone bosons to become the longitudinal modes of the W ± and Z bosons. The pseudoscalar Higgs boson A 0 and charged Higgs boson H ± are mass eigenstates with masses m A and m H + , respectively. We also define the mass eigenstates of the CP-even neutral Higgs bosons as h and H, with respective masses m h and m H (m h < m H ), through a rotation of angle α from the original PQ basis as follows: Note that the light Higgs boson h becomes a SM-like Higgs boson h SM in the limit of sin(β − α) → 1, which can be realized when m 2 12 → ∞. The couplings between h and weak gauge bosons are read as where g SM hV V are the couplings in the SM. The triple Higgs coupling λ hH + H − , defined by the λ hH + H − hH + H − interaction term in the Lagrangian, is given by: where m 2 A ≡ 2m 2 12 / sin 2β. For Yukawa interactions, since we consider the scenario in which only the right-handed top quark carries a nonzero PQ charge among all quark fields, the up-type Yukawa interaction Lagrangian is: where the family indices a = 1, 2 and i, j = 1, 2, 3. Schematically, the Yukawa coupling matrices, Y u1 and Y u2 , in the original PQ basis take the forms: where * indicates a generally nonzero element.
In the Higgs basis, the up-type Yukawa interaction Lagrangian can be expressed as and the Yukawa matrices are Therefore, the number of degrees of freedom in the up-type Yukawa matrices of this model is the same as that of the SM. At this stage, the mass matrix is generally non-diagonal, and can be brought to its diagonal form through a bi-unitary transformation where U and V are two unitary matrices, which rotates the left-handed fields q i and the right-handed fields u R,i , respectively. In this basis, the other Yukawa matrix becomes where the Hermitian matrix Note that in the second term of Eq. (10), the (tan β + cot β)H u part describes mixing among up-type quarks and Y diag u controls the strength of coupling with the dominant component given by the top Yukawa coupling. For simplicity, we will omit the superscript "diag" while working in the mass-diagonal basis in the following discussions. Note that V is the rotation matrix for the right-handed up-type quarks and is completely independent of the CKM matrix, which is the product of left-handed up-type and left-handed down-type quark rotation matrices. Therefore, the mixing angles in V can be as large as O(1), a key intriguing feature of the model.
As an illustration and in anticipation of interesting collider phenomenology associated with the top quark, we restrict ourselves to the case of t-c mixing in this paper as in Ref. [14].
In such a simplified scenario without introducing new CP phases, the mixing matrices H u and V can be parameterized in terms of ρ only as: and the Yukawa interactions of the observed Higgs boson h in the mass eigenbasis are then described by where a ≡ (tan β + cot β) cos(β − α) and, One striking feature of L FCNC is that the predicted flavor violation is associated with large asymmetries in the chirality. In this simplified case, the top decay is dominated by the right-handed charm-associated processes to be discussed in more detail in the next section.

A. Rare Top quark FCNC decay
This model generically predicts the top FCNC decay t → ch (or t → uh) via the mixing effect. Therefore, such decays serve as a smoking gun signature of the model. For definite-ness, we focus on the t → ch decay in this section, but note that the current experimental limits do not actively tag the flavor of the accompanied jet and that what is constrained is the weighted sum of all branching ratios of t → qh (q = u, c).
The partial decay width of t → ch is given by with r 2 h ≡ m 2 h /m 2 t ∼ 0.522 for m h = 125 GeV and m t = 173 GeV. By comparing it with the width of t → bW in the SM at the leading order, .214 for m W = 80.4 GeV, we can obtain by assuming BR(t → bW ) is close to unity that The nominal branching ratio of 0.22 % corresponds to the mixing parameters satisfying a 2 sin 2 ρ = 0.068 .
As the future sensitivity of 0.02% for BR(t → ch) (for 14 TeV and 3000 fb −1 ) corresponds to a 2 sin 2 ρ = 6.2 × 10 −3 , the current nominal value will be fully confirmed at 5σ level by then if the model is correct.
These two values will be depicted in the following figures in pink and red, respectively.
It is noted that the h-t-c/u couplings can also contribute to other flavor observables.
While introducing no new CP-violating sources in this model, we note in passing that the imaginary parts of the flavor-violating Yukawa couplings can be constrained by the hadron electric dipole moments and CP-violating observables in the D mesons [28].

B. Perturbativity constraints
Since Y ,diag u involves ρ and tan β, some element may become too large for some sizeable ρ, tan β or cot β. Large Yukawa couplings could have the problem of blowing up after running to high energies. Since our model assumes the PQ symmetry to solve the strong CP problem, the coupling must not blow up at least up to the PQ scale. More stringently, we require that the theory does not contain any divergent coupling up to the Planck scale. As the Yukawa couplings are base-dependent, we require that the absolute value of any Yukawa coupling is smaller than 4π for the validity of perturbation. We use the 1-loop renormalization equations in Eqs. (404) to (409) of Ref. [29], ignoring the Higgs self couplings, for numerical evaluations. The parameter region satisfying the above perturbative condition is shown in Fig. 1, with the excluded region shown in gray.

III. HIGGS SIGNAL STRENGTH CONSTRAINTS FROM LHC RUN-II
As we assume that the exotic Higgs bosons are sufficiently heavy to decouple from the lowenergy phenomenology [14], our model is essentially parametrized by only three parameters α, β and ρ. In this section, we show the constraints on these model parameters using the latest LHC Higgs data. As noted earlier, the couplings between the SM-like Higgs boson h and the SM particles are modified from their SM values: Eq. (4) for the gauge bosons and Eq. (15) for the fermions. We use them to estimate the signal strengths of various Higgs production channels. We note that the diphoton decay width depends to some extent on the coupling λ hH + H − , which in turn would modify the predicted Higgs signal strengths.
However, such a dependence diminishes under our assumption of heavy exotic Higgs bosons.
Therefore, for definiteness, we set λ hH + H − = 0 in the following analysis.
In Fig. 2, we show the allowed parameter space in the (a, ρ) plane for tan β fixed to 0.5, 1, 2 and 10. The darker blue, blue, and lighter blue regions correspond respectively to the 1σ, 2σ and 3σ regions, based on ∆χ 2 = χ 2 − χ 2 min = (3.53, 8.02, 14.2) for 3 degrees of freedom. We found χ 2 min = 48.3 being an appropriate goodness of fit for 44 observables. For a fixed tan β value, the blue regions always become broader along the a direction when ρ is turned on and broadest around π/2 to 3π/4. This tendency is stronger when tan β is smaller. This can be understood as follows from the fact that the latest Higgs data are essentially consistent with the SM expectations, and a large deviation in the signal strengths disfavored. The gauge boson couplings to the Higgs are consistent with the SM, and it forces sin(β − α) 1 and cos(β − α) to be small. Thus, each signal strength µ i from the ggHinitiated mode is roughly proportional to (ξ t /ξ b ) 2 , taking the fact that the total width is mainly controlled by the bottom Yukawa coupling. From the expression Eq.(15), we see d(ξ t /ξ b ) (1 + cos ρ)(tan β + cot β)d cos(β − α), and this part of the corrections becomes milder with non-zero ρ. Moreover, for small tan β, a larger a is realized due to large cot β whereas it does not initiate large effects on ξ b , or on the total width.
There is also so-called "wrong-sign Yukawa" solution with ξ t = 1 and ξ i = −1 (i: other than the top) for large tan β. In this case, the Yukawa couplings of quarks other than the top quark have an opposite sign to their SM values [46], achieved by having tan β cos(β−α) −2 but (1 − cos ρ) tan β cos(β − α) ∼ 0. One can understand why the corresponding solution does not exist for small tan β nor for large ρ from this expression. For tan β = 10, there remains viable parameter space at a ∼ 2.2 as well as normal a ∼ 0 region for small ρ although the wrong-sign region is compatible with the Higgs data at the 2σ level, as seen in the right lower plot of Fig. 2.
The FCNC top rare decay process t → ch can put useful constraints on the parameter space as well. Such FCNC effects are proportional to a 2 sin 2 ρ. The nominal FCNC branching ratios of ≤ 0.22 % and ≤ 0.02% are depicted in the plots by the pink and red regions, respectively. If the nominal size of the signature top decay turns out to be a real signal, we can exclude ρ ∼ 0 region allowed by the current Higgs signal strength data.
In Fig. 2, we also superimpose the requirement of perturbativity in the Yukawa couplings.
For tan β > 1 (tan β < 1), larger (smaller) ρ regions are ruled out, as indicated by the shaded light-gray region. There is no such a constraint for the tan β = 1 case. However, we have a dark-gray region ruled out by c βα > 1 in this case.  In the case with the mixing effect is larger, the allowed parameter region is slightly relaxed but still constrained to be |a| < ∼ 0.3 for tan β > 1. On the other hand, for the smaller tan β region, the allowed a region is significantly extended. Furthermore, one can see in the right plot that even with ρ = 3π/4, which provides almost maximized FCNC's in ρ, larger tan β is incompatible with BR(t → ch) = 0.22%. Therefore, we can conclude that there exists an upper bound on tan β for the nominal value of BR(t → ch) = 0.22% 1 . Larger tan β with larger mixing ρ is also disfavored by the perturbativity requirement.

IV. FURTHER TESTS FOR THE MODEL
Once we observe a sufficient number of t → ch events, it will be possible to check the chiral nature of the Higgs flavor-changing couplings as predicted in the model: the charm quark in the decay product should be right-handed. The spin analyzing power κ i of particle i in the decay product is defined as where P is the polarization of the mother particle along a specific direction, called the polarization axis, Γ i is the partial decay width of the mode containing particle i, and θ i is the polar angle of particle i with respect to the polarization axis. The spin analyzing power of the charged lepton, κ + , from the usual top decay t → b + ν is known to have the largest value +1 at leading order [47]. We denote the spin analyzing power for the anti-top quark decay byκ, and note thatκf = −κ f assuming CP invariance. Our model predicts dΓ t→ch /d cos θ ∝ 1+cos θ, and the charm quark and the Higgs boson have the spin analyzing powers κ c = +1 and κ h = −1, respectively. Once we know the original top spin direction, we can readily determine κ c and κ h in the t → ch decay.
We have discussed in Ref. [14] the possibility of determining the chirality structure using the top spin correlation in tt production at the LHC [48,49] as the top quarks in top pair production are not polarized and not directly usable. The differential cross section in the double theta distribution is given by: where θ i,j are defined in the rest frame of the top and anti-top quarks, respectively, and the tt spin asymmetry defined in the helicity basis [50] is at the LHC.
For a rough estimate of the required number of events to determine κ h (or κ c ) by measuring the angular distribution of i = + , j = h and the corresponding anti-particle case, we have also introduced a simpler observable out of the above-mentioned observables as To confirm that κ h ∼ −1, we have to measure a positive A h at a precision better than 0.088. The statistical uncertainty on A h is then given by implying that we need at least ∼ 130 signal events to confirm the decay distribution structure at the 1σ level. As we expect 3 × 10 9 top pair events using σ(tt) ∼ 1 nb and an integrated luminosity of 3000 fb −1 at the 14-TeV LHC, it provides ∼ 10 7 t → ch events for the nominal branching ratio of 0.22%. Even considering only the cleanest mode h → γγ, we still expect ∼ 5000 events after multiplying BR(h → γγ) ∼ 2.3 × 10 −3 and the leptonic decay branching ratio of the top quark. Besides, the h → bb mode can be incorporated to enhance the signal significance [51]. Therefore, with the assumption of the nominal branching ratio, one can easily determine the chirality structure of the flavor-changing Higgs coupling at the ultimate integrated luminosity in LHC Run-III.
Recently, the high energy upgrade of the LHC (HE-LHC) is more sceriously discussed and its target center of energy and integrated luminosity are realistically decided as 27 TeV and 12 fb −1 [52]. At √ s = 27 TeV, the tt cross section reaches 3.7 nb computed by Hathor 2.0 [53], therefore, we expect that the sensitivity on the branching ratio improves below 10 −5 due to 15 times as many as tt events.

V. CONCLUSION
In this Letter, we consider the current status of the top-specific variant axion model in light of the latest Higgs data and a slight excess in the t → ch decay . This model is well-motivated to solve the strong CP and domain wall problems. As the top FCNC decay is one of the generic predictions of the model, we discuss whether it is possible to have a sizeable BR(t → ch) under various theoretical and phenomenological constraints. We have found that to realize a rather large branching ratio of 0.22% in this model, tan β ∼ 1 or smaller is preferred by the current Higgs data. Such a preference is also supported by the perturbativity requirement on the Yukawa couplings. In other words, what we have shown is that although the Higgs signal strength data are essentially the same as the SM predictions, our model can readily accommodate a sizable BR(t → ch) without conflicts with the Higgs data as long as tan β < ∼ 1.
The h-t-c vertex has a specific chirality structure according to the model. We therefore propose to measure this characteristic feature as an essential step toward verifying the model.
We have shown that this can be done by measuring the spin correlation in the top pair production through the t → ch mode and estimated that the required sensitivity for confirming the model can be achieved by the end of LHC Run-III, assuming BR(t → ch) = 0.22% is realized in Nature.