Probing for Extra Top Yukawa Couplings in Light of $t\bar th(125)$ Observation

The observation of $t\bar th(125)$ production at the Large Hadron Collider (LHC) is the first direct measurement of the top Yukawa coupling. It opens the window on an extra top Yukawa coupling, $\rho_{tt}$, from a second Higgs doublet, without a $Z_2$ symmetry to forbid flavor changing neutral Higgs couplings. We show that $t\bar th$ and Higgs property measurements at the High Luminosity LHC can constrain the ${\rm Re}\,\rho_{tt}$--${\rm Im}\,\rho_{tt}$ parameter space that could drive electroweak baryogenesis, but the $\Gamma_h$ width measurement must be considerably improved beyond current projections.

µ tth = 1.32 +0. 28 −0.26 , (Run 2 Update, ATLAS) (2) at 5.8σ significance (4.9σ), while combining further with Run 1 data, the significance becomes 6.3σ (5.1σ). We have put the SM expectation in parenthesis. Thus, both ATLAS and CMS have observed tth production. We note that both experiments had earlier hints for tth production with strength stronger than SM, which improve the combined significance quoted above.
Since Yukawa couplings are the source of CP violation (CPV) in SM, with direct measurement of top Yukawa coupling attained, it opens up the question whether there are additional Yukawa couplings. As there is nothing against the existence of a second doublet, the two Higgs doublet model (2HDM) is in fact one of the most plausible beyond-SM (BSM) possibilities, where one should "naturally" have a second set of Yukawa couplings. However, due to the perceived curse of flavor changing neutral Higgs (FCNH) couplings, Glashow and Weinberg famously eliminated all such extra couplings by demanding "natural" flavor conservation (NFC) [4]: each type of fermion charge receives mass from just one Higgs doublet. This enforces only one Yukawa matrix per fermion mass matrix, and they are therefore simultaneously diagonalized: absence of FCNH. There are two ways to implement the NFC condition for quarks, and 2HDM-II, where u-type and d-type quarks receive mass from separate doublets, has been the most popular, as it arises with supersymmetry.
A decade after the Glashow-Weinberg NFC condition, the emerging quark mass-mixing hierarchy led to the critique [5] that NFC may be overkill. As the top quark is the heaviest fermion, the best probe may be t → ch [6] for FCNH tch coupling. With the observation of h(125) in 2012, it was stressed [7] that the 2 × 2 extra Yukawa couplings ρ cc , ρ ct , ρ tc , and ρ tt of the exotic doublet should be taken seriously, and the issue is experimental: we have to demonstrate their nonexistence, rather than assume NFC and throw them away.
The tth coupling in 2HDM without NFC is while the tch coupling is (ρ ct is already constrained by flavor physics to be small [7,8]) where cos 2 γ ≪ 1 is the alignment phenomenon observed at the LHC [9], the fact that the observed h boson is rather close [10] to the SM Higgs boson. In 2HDM-II, the mixing angle of the two CP -even scalars, cos γ, is usually expressed as cos(α − β), but without the NFC condition, or the Z 2 symmetry to implement it, tan β is  unphysical, hence we use the different notation [11]. It was recently noted that λ t Im ρ tt can easily drive [12] the baryon asymmetry of our Universe (BAU), which needs O(1) Higgs quartic couplings of the 2HDM for the first order electroweak phase transition. The latter can relatively easily accommodate [11] the observed approximate alignment phenomenon. These add to the attraction of 2HDM without the NFC condition. So far, the t → ch decay has not been observed, with limits approaching 10 −3 [13]. We assume it is suppressed by ρ tc , but take the maximum cos γ value allowed by data. The point is, even if the admixture of ρ tt into the tth coupling is suppressed by approximate alignment, i.e. cos γ, interference with the leading SM λ t effect provides a sensitive probe in tth production. In the following, we illustrate the new, direct probe of tth production, and compare with indirect probes of h → gg, γγ loop processes, projecting into the future of High-Luminosity LHC (HL-LHC). We give a simplified discussion of how other extra Yukawa couplings, such as ρ bb , would affect Γ h and shift the loop constraint. We offer some remarks on future prospects beyond HL-LHC.
Indirect and Direct Probes of ρ tt .-ρ tt and cos γ are constrained by ATLAS and CMS measurements of the Higgs boson production and decay rates. The main effect of ρ tt is for gg → h, which is given by [14,15] through triangle loop diagram, where the absorptive, i.e. explicit i terms arise from light quark loops. The (Re ρ tt ) 2 effect is suppressed by (cos γ) 2 , or alignment, where we shall take the value of cos γ = 0.3 (correspond-ing to − sin γ = 0.954) that may still be allowed by data [8]. Of interest is the sin γ cos γ interference term between the SM and extra Yukawa coupling, which allows better sensitivity to Re ρ tt . Although the (Im ρ tt ) 2 term is also suppressed by (cos γ) 2 , but because the imaginary part of the extra Yukawa coupling leads to a γ 5 coupling, the term receives a numerical factor of order 2.6 (see e.g. Ref. [16] for discussion of pseudoscalar coupling), making gg → h sensitive to larger values of (Im ρ tt ) 2 .
The h → W W * , ZZ * rates are modified by the overall factor sin 2 γ from the SM ones. As for h → γγ decay, which also arises from triangle loop, the rate is only mildly affected [14,15] by cos γ ρ tt because of W boson dominance in the loop. Using the Run 1 combination of ATLAS and CMS results [10], constraints on real and imaginary parts of ρ tt for cos γ = 0.3 with sin γ < 0 are shown in Fig. 1. We use the ten-parameter fit to the three decay channels h → γγ, ZZ * , W W * with ggF+ttH or VBF+V H production, and individual 2σ constraints from the six signal strengths are overlaid, resulting in the gray shaded regions. The right-hand side (r.h.s.) comes from the h → W W * decay mode, while l.h.s. comes from h → ZZ * . The actual sensitivity to Re ρ tt and Im ρ tt is mainly driven by gg → h production.
For the direct probe of tth coupling, we calculate the pp → tth cross section at leading order (LO) by Monte Carlo event generator MadGraph5 aMC@NLO [17] with the parton distribution function set NN23LO1 [18]. In particular, we use the Higgs Characterisation model [19] implemented in FeynRules 2.0 [20] framework, where model details can be found in Ref. [21]. We ignored contributions other than from tth coupling. The signal strength for 13 TeV LHC can be approximated by with mild modification of the 0.45 coefficient to 0.46 for 14 TeV LHC. Thus, the (Re ρ tt ) 2 and (Im ρ tt ) 2 terms are suppressed by | cos γ| 2 , but the sin γ cos γ interference term brings in better sensitivity to Re ρ tt . We take the ATLAS Run 2 update of µ tth , Eq.
(2), and display the 2σ allowed range in Fig. 1[left] as marked. Because of the mild excess, a positive Re ρ tt is preferred, but partially excluded by the indirect data. However, a broad range of |Im ρ tt | is allowed, extending beyond 1 if the Re ρ tt interference effect is destructive. Thus, current data between indirect and direct probes of ρ tt are quite consistent with electroweak baryogenesis [12] by 2HDM without NFC.
It is of interest, therefore, to project the reach for HL-LHC. This is shown as the blue and red lines in Fig. 1[right] for µ tth = 0.81, 1.32, corresponding to −2σ and central value in Eq. (2), respectively. Considering the trend in measurements of µ tth , we do not display the +2σ case. On the other hand, we take an optimistic 5% as the 1σ uncertainty reach for ultimate HL-LHC sensitivity, based on current projections [22], and anticipating a combination of ATLAS and CMS data. However, we have left the indirect probes, the shaded region, unchanged from the Run 1 ATLAS-CMS combination, as it is rather hard to estimate the HL-LHC sensitivity reach. Although Run 2 analyses are available from ATLAS and CMS, the results so far [23] suggest correlations between the values of ggF (gluon-gluon fusion) and VBF (vector boson fusion) production, and in any case the two datasets are not yet combined. A full Run 2 combina-tion is probably a couple of years away. We also note that, as data increases, separate production/decay channels would likely be disentangled.
But one can visualize a narrower "allowed white crescent" from the indirect measurements. Depending on the overlap with the µ tth band, the allowed region could be a specific Re ρ tt with restricted Im ρ tt range, or more interestingly, a preference for relatively large values of Im ρ tt . Either way, without probing CPV directly, this would provide insight on the origin of BAU.
Simplified Effect of Light Fermions.-But there is a catch. Analogous to Eq. (3), the 2HDM without NFC brings in extra Yukawa couplings that modify hbb, hτ τ and hcc couplings. While they give minor modifications to gg → h and h → γγ, the major impact is on the total h width, Γ h , which is not well measured yet.
As individual processes are also not yet well measured, rather than several couplings, we treat the partial width of h → light fermions as the single overall effect, Taking branching ratio values [24] for m h = 125.09 GeV, the total width is modified as Similar to Fig. 1[right], we display the effect of Γ all f f = 0.5 (1.5)Γ SM all f f in Fig. 2[left] ([right]), where the "allowed white crescent" is moved leftward (rightward). We remark that 2HDM without NFC can relatively easily lead to Γ h that is narrower than in SM. We note in passing that the constraint on r.h.s. of Fig. 2[right] now arises from VBF production with h → W W * decay.
The µ tth ∼ 0.81 band in Fig. 2[left] illustrates the situation where Re ρ tt is slightly negative, with |Im ρ tt | up to 1 fully allowed, while the µ tth ∼ 1.32 band illustrates a tension between indirect and direct probes of ρ tt . The µ tth ∼ 0.81 band in Fig. 2[right] illustrates the situation where large Im ρ tt is indicated, while the µ tth ∼ 1.32 band illustrates constructive interference, but a broad range of Im ρ tt is allowed.
However, since the "white crescent" would become narrower with HL-LHC data, what Fig. 2 really shows is that a good measurement of Γ h is needed [25]. With current projections at 50% of Γ SM h [26] for HL-LHC, we can only hope that δΓ h can be much improved with actual data, otherwise we would not really know what is the overlap region, as 2HDM without NFC would shift all diagonal Yukawa couplings, in principle by same order as the corresponding SM Yukawa coupling, modulated by cos γ. More precise measurements of µ cc , µ τ + τ − and especially µ bb may help. Another approach, for example, is VBF production followed by h → V V * , which could probe Γ h through branching ratio measurement, as the V V * couplings are not much affected by cos γ ρ tt .
Discussion and Conclusion.-We stress that 2HDM without NFC offer new Yukawa couplings that could alter all f f h couplings from SM values, modulated by cos γ. This makes clear the importance of a complete program to measure µ bb , µ τ + τ − , µ µ + µ − , and even µ cc if charm tagging could be vastly improved.
A second point to note is that each one of these diagonal Yukawa coupling corrections are generally complex. For example, Im(ρ tt ) contributes to the electron EDM through two-loop contributions. Under the assumption that the electron Yukawa coupling is SM-like (ρ ee = 0) and heavy scalar contributions are negligible, the recent ACME result [27] would imply | cos γ Im(ρ tt )/λ t | < 0.01 [12,14]. However, allowing for a complex, in particular imaginary, ρ ee with strength similar to λ e of SM, it can in principle induce cancellation [12] of the two-loop effect. We had tacitly assumed this in exploring tth, and illustrates how the 2HDM without NFC could affect flavor physics. Note that, given that λ t ∼ = 1 is already known, the dominant eigenvalue of the other combination of the two u-type Yukawa matrices, viz. ρ tt , is likely O(λ t ) hence O(1), and with phase arbitrary. Similar argument would hold for ρ bb and ρ τ τ .
In contrast to usual effective Lagrangian discussions, the interaction terms reflected in Eqs. (3) and (4) are as fundamental as the Yukawa couplings in SM. On one hand, they are well hidden by approximate alignment, or the smallness of cos γ. Thus, the direct pp → tth and other indirect probes would rapidly weaken for smaller cos γ. Furthermore, as outlined in Ref. [11], the massmixing hierarchy, or some flavor-organization principle, could control FCNH involving lighter generations, and together with approximate alignment, can fully replace the NFC condition to explain the absence of low energy FCNH effects. But alignment need not [11] imply decoupling, and the exotic Higgs sector could well be sub-TeV in mass. If alignment is effective, one would have to probe this exotic Higgs sector, for example via cg → tH 0 , tA 0 production, leading to novel ttc (same-sign top) and ttt (triple top) signatures [28] at the LHC.
With the hope that alignment does not work too well, i.e. cos γ is not overly small, what is the future outlook? First, cos γ should be studied more generally, free of the Z 2 symmetry mindset (i.e. beyond Ref. [9]). A measurement of, rather than constraint on, cos γ would be astounding. Second, improved projections for HL-LHC is expected with a CERN Yellow Report that is under preparation for the European Particle Physics Strategy Update. But the projections must be continuously updated as experience is gained with larger datasets, including on Γ h measurement. Third, we are at the juncture of ILC(250) [29] or CLIC(380) [30] decision. Although these machines are still far away, they provide great hope for much more precision in Higgs property measurements, which would provide better indirect constraints, including on δΓ h . However, tth production would require at least 500 GeV e + e − collision energy.
Thus, the high energy extension of LHC looks more promising for the nearer future on direct tth probe. There is no doubt that a 100 TeV machine [31], though much farther away, would advance the tth and exotic Higgs frontiers by great stride. If the alignment phenomenon reflects [11] O(1) Higgs quartic couplings within 2HDM, there is likely another layer of BSM physics at the 10 TeV scale to be explored. Finally, we have advocated simple probes of just measuring rates. More sophisticated angular or asymmetry analyses [32] can probe the CPV nature of the tth coupling directly.
In conclusion, the observation of pp → tth 0 at LHC is the first direct measurement of the top Yukawa coupling, and offers a window on the extra Yukawa coupling ρ tt from a second Higgs doublet where the NFC condition is not imposed. The large λ t ∼ = 1 of SM provides the lever arm to probe cos γ ρ tt through interference, where cos γ is the CP -even Higgs mixing angle. The parameter space for electroweak baryogenesis offered by this 2HDM can be probed by CP -conserving tth 0 production rate and Higgs property measurements. The Achilles heel for this program at the HL-LHC is the knowledge of the h 0 width, Γ h 0 , and ATLAS and CMS should put a premium on its improved measurement.