In this section we will show how it is possible to promote the perturbative unitarity obtained from the one-loop computations to the exact (in a sense that will be clarified below) unitarity. To show how that can be done we will start first with the particular case \(a^2=b\). From the results in [26, 27] it is very easy to realize that the \(\omega \omega \rightarrow hh\) amplitude vanishes, and hh decouples from \(\omega \omega \) elastic scattering. Thus, the reaction matrix can be written in \(2\times 2\) form as
$$\begin{aligned} F = \begin{pmatrix} A_{00} &{} Q_0 \\ Q_0 &{} S_0 \end{pmatrix} \equiv \begin{pmatrix} A &{} Q \\ Q &{} S \end{pmatrix}, \end{aligned}$$
(80)
where \(A_{00}\), \(Q_0\) and \(S_0\) are the elastic \(\omega \omega \rightarrow \omega \omega \), cross-channel \(\omega \omega \rightarrow t\bar{t}\), and elastic \(t\bar{t}\rightarrow t\bar{t}\)
\(I=J=0\) partial waves, respectively. As we saw in the previous section the A, Q and S amplitudes can be expanded as
$$\begin{aligned} A&= A^{(0)} + A^{(1)} + \cdots&\sim 1 ,\end{aligned}$$
(81a)
$$\begin{aligned} Q&= Q^{(0)} + Q^{(1)} + \cdots&\sim \mathscr {O}\left( \frac{M_t}{v}\right) ,\end{aligned}$$
(81b)
$$\begin{aligned} S&= S^{(0)} + \cdots&\sim \mathscr {O}\left( \frac{M_t^2}{v^2}\right) . \end{aligned}$$
(81c)
On the (RC) right cut (the physical region) the unitarity relation \(\mathrm {Im}F = F F^\dagger \) applies, entailing
$$\begin{aligned} \mathrm {Im}A&= |A |^2 + \mathscr {O}\left( \frac{M_t^2}{v^2}\right) ,\end{aligned}$$
(82a)
$$\begin{aligned} \mathrm {Im}Q&= A Q^* + \mathscr {O}\left( \frac{M_t^3}{v^3}\right) ,\end{aligned}$$
(82b)
$$\begin{aligned} \mathrm {Im}S&= 0 + \mathscr {O}\left( \frac{M_t^2}{v^2}\right) . \end{aligned}$$
(82c)
These equations, if expanded in perturbation theory, return Eqs. (78a), (78d) and (78f) (setting \(M=0\) there). In order to fulfill these relations we proceed as follows. First, we consider the elastic scattering \(\omega \omega \) amplitude A. As shown in [26, 27], Eq. (78d) can be satisfied by using the inverse amplitude method (IAM), which introduces the unitarized amplitude
$$\begin{aligned} \tilde{A}=\frac{(A^{(0)})^2}{A^{(0)}-A^{(1)}}. \end{aligned}$$
(83)
This ensures elastic unitarity, \(\mathrm {Im}\tilde{A} = \tilde{A}\tilde{A}^*\), provided we have perturbative unitarity, i.e. \(\mathrm {Im}A^{(1)} = (A^{(0)})^2\), as is the case here. Now, in order to unitarize Q we introduce
$$\begin{aligned} \tilde{Q} = Q^{(0)} + Q^{(1)}\frac{\tilde{A}}{A^{(0)}} . \end{aligned}$$
(84)
Again, it is very easy to show that this partial wave fulfills the unitarity relation \(\mathrm {Im}\tilde{Q} = A Q^*\) by using the perturbative result \(\mathrm {Im}Q^{(1)}=Q^{(0)}A^{(0)}\) as follows:
$$\begin{aligned} {\tilde{Q}} \big |_{\mathrm{RC}}&= Q^{(0)} + Q^{(1)}\frac{{\tilde{A}}}{A^{(0)}} \nonumber \\&= \left[ Q^{(0)}\left( 1-\frac{A^{(1)}}{A^{(0)}}\right) + Q^{(1)}\right] \frac{A^{(0)}}{A^{(0)}-A^{(1)}} \nonumber \\&= \left[ Q^{(0)} - \frac{Q^{(0)}}{A^{(0)}}\mathrm{Re}A^{(1)} + \mathrm{Re}Q^{(1)} \right] \frac{{\tilde{A}}}{A^{(0)}}. \end{aligned}$$
(85)
Thus, we have
$$\begin{aligned} \mathrm {Im}{\tilde{Q}} \big |_{\mathrm{RC}}= & {} \left[ Q^{(0)} - \frac{Q^{(0)}}{A^{(0)}}\mathrm{Re}A^{(1)} + \mathrm{Re}Q^{(1)} \right] \frac{\mathrm {Im}{\tilde{A}}}{A^{(0)}} \nonumber \\= & {} \left[ Q^{(0)} - \frac{Q^{(0)}}{A^{(0)}}\mathrm{Re}A^{(1)} + \mathrm{Re}Q^{(1)} \right] \frac{{\tilde{A}}{\tilde{A}}^*}{A^{(0)}} = {\tilde{Q}}{\tilde{A}}^{*}.\nonumber \\ \end{aligned}$$
(86)
Hence, we recover Eq. (82b) as announced. Therefore we are left with two unitarized amplitudes \(\tilde{A}\) and \(\tilde{Q}\) for the processes \(\omega \omega \rightarrow \omega \omega \) and \(\omega \omega \rightarrow t\bar{t}\), respectively. These amplitudes also respect the perturbative expansion
$$\begin{aligned} \tilde{A}&= A^{(0)} + A^{(1)} + \cdots ,\nonumber \\ \tilde{Q}&= Q^{(0)} + Q^{(1)} + \cdots , \end{aligned}$$
(87)
and, in addition they feature the proper analytical structure on the whole complex plane. In particular they have a right (unitarity) cut and also the expected left cut. Moreover, they can be analytically extended to the second Riemann sheet beyond the unitarity cut and they can have poles there that can be understood as resonances (whether “dynamic” or “intrinsic”) developing in some regions of the chiral coupling space. Those resonances are typical of strongly interacting scenarios for the symmetry-breaking sector of the SM and are under active research at the LHC.
The unitarity condition for the Q amplitude linking \(\omega \omega \) and \(t\bar{t}\) introduced in Eq. (84) can now be checked numerically, and we have done so (not shown). Our numeric precision is, for the entire energy interval of interest up to \(3\,\mathrm{TeV}\), of order \(10^{-5}\) without any particular effort (and this small error probably stems from our setting b not quite equal to \(a^2\) to avoid numerical problems elsewhere, so that a tiny leak to the hh channel may be present), so that Watson’s final state theorem is well satisfied and the phase of the Q amplitude is correctly set to that of the strongly interacting \(A(\omega \omega \rightarrow \omega \omega )\).
We now exemplify the power of a method by generating a resonance in the elastic \(A(\omega \omega \rightarrow \omega \omega )\) amplitude and feeding it to the \(t\bar{t}\) channel. We choose \(\mu \simeq M=750\)GeV, so that a comparison with many other theory work can be made, that was inspired by a presumed narrow LHC excess at around 0.75 TeV, \(a=0.81\), \(a_5=0.0023\), and all other parameters from the \(\omega \omega \) sector as in the SM (particularly \(a_4=0\) and \(b\simeq a^2\)). This generates a relatively narrow resonance with mass around 750 GeV, \(\Gamma /M\simeq 0.06\) similar to what the community was considering before it was clear that it had been a statistical fluctuation. The resonance is shown in the top plot of Fig. 10. The lower plot shows its effect on the real part of the Q amplitude, where we have set the parameter \(c_1\) to \(\pm 1\), differently from zero.
As can be seen in the figure, the resonance, a textbook Breit–Wigner resonance in the A elastic channel (EWSBS) appears as a dip due to its interference with the background in the \(Q(\omega \omega \rightarrow t\bar{t})\) amplitude. Of course, such dips will appear broadened and lessened after convolution with the parton distribution functions producing the top–antitop system, the hard kernels, and the reconstruction efficiency (and detector acceptance) of the final product decays. Though a full simulation is beyond the scope of this work, it is possible that they are observable, providing a signal that is not so often expected (as practitioners often seek excess cross sections). A similar phenomenon has been observed by [34] in interference between perturbative SM production \(WW\rightarrow t\bar{t}\) (the equivalent of our Q amplitude) and the s-channel production of \(t\bar{t}\) via a new resonance of \(\mathscr {O}\)(TeV) mass. The coincidence suggests that this may be a robust result. These interference phenomena of backgrounds and narrow resonances are well known in hadron physics [54, 55] and it would be interesting to discover them in the EWSBS.
The entire discussion can now be extended to the more general case \(a^2\ne b\) where the cross-channels \(\omega \omega \rightarrow hh\) and \(hh\rightarrow t\bar{t}\) are active again. Then the reaction matrix is \(3\times 3\),
$$\begin{aligned} F = \begin{pmatrix} A_{00} &{} M_0 &{} Q \\ M_0 &{} T_0 &{} N \\ Q &{} N &{} S \end{pmatrix} \equiv \begin{pmatrix} A &{} M &{} Q \\ M &{} T &{} N \\ Q &{} N &{} S \end{pmatrix} . \end{aligned}$$
(88)
Here, \(A_{00}\), \(Q_0\) and \(S_0\) are the scalar partial waves as in Eq. (81a) and following. Now, since the crossed channel amplitude \(\omega \omega \rightarrow hh\) is present, we also include its \(J=0\) partial wave, and those for \(hh\rightarrow hh\) and \(hh\rightarrow t\bar{t}\). All of them accept a chiral expansion as
$$\begin{aligned} F = F^{(0)} + F^{(1)} + \dots , \quad \mathrm {Im}F^{(0)} \equiv 0 . \end{aligned}$$
(89)
Once again, the partial waves Q, N and S are suppressed by \(M_t/v\) factors. In particular,
$$\begin{aligned} X&= X^{(0)} + X^{(1)} + \cdots\sim & {} 1 ,\end{aligned}$$
(90)
$$\begin{aligned} Q&= Q^{(0)} + Q^{(1)} + \cdots\sim & {} \mathscr {O}\left( \frac{M_t}{v}\right) ,\end{aligned}$$
(91)
$$\begin{aligned} N&= N^{(0)} + N^{(1)} + \cdots\sim & {} \mathscr {O}\left( \frac{M_t}{v}\right) ,\end{aligned}$$
(92)
$$\begin{aligned} S&= S^{(0)} + \cdots\sim & {} \mathscr {O}\left( \frac{M_t^2}{v^2}\right) , \end{aligned}$$
(93)
where \(X= A\), M or T. On the RC, the unitarity relation \(\mathrm {Im}F = F F^\dagger \) applies, which leads to the set of Eq. (75), where we omit terms suppressed by higher powers of \(M_t/v\) in such a way that all equations are correct up to \(\mathscr {O}(M_t^2/v^2)\). This is essential to be able to decouple the unitarization of the WBGBs sector (the A, M and T partial waves) from the \(t\bar{t}\) amplitudes. For the unitarization of the WBGBs sector we can use again the (coupled) IAM method. For this purpose, we first define the \(2\times 2\) matrix
$$\begin{aligned} K \equiv \begin{pmatrix} A &{} M \\ M &{} T \end{pmatrix}, \end{aligned}$$
(94)
which as usual admits a chiral expansion \(K=K^{(0)}+K^{(1)}+\cdots \) with \(\mathrm {Im}K^{(1)}= K^{(0)}K^{(0)}\) on the RC (perturbative unitarity). The corresponding unitarized matrix \(\tilde{K}\) is provided by the IAM method (basically, a dispersive analysis for this matrix that employs the chiral expansion on the LC and everywhere for small s, and exact two-channel unitarity on the RC), that generalizes Eq. (83)
$$\begin{aligned} \tilde{K}=K^{(0)}(K^{(0)}-K^{(1)})^{-1} K^{(0)} \ . \end{aligned}$$
(95)
By construction, the unitarity relation \(\mathrm {Im}\tilde{K} = \tilde{K}\tilde{K}^\dagger \) holds. Now, the remaining unitarity conditions on the RC, Eqs. (75d)–(75f), can be written in a condensed way as
$$\begin{aligned} \mathrm {Im}\begin{pmatrix} Q \\ N \end{pmatrix}= K \begin{pmatrix} Q \\ N \end{pmatrix}^*. \end{aligned}$$
(96)
This can also be expanded in perturbation theory,
$$\begin{aligned} \mathrm {Im}\begin{pmatrix} Q^{(1)} \\ N^{(1)} \end{pmatrix}= K^{(0)}\begin{pmatrix} Q^{(0)} \\ N^{(0)} \end{pmatrix}. \end{aligned}$$
(97)
A solution to Eq. (96) can be written down generalizing the simpler \(a^2= b\) discussion. The unitarized amplitudes are then (a simple demonstration is relegated to the appendix)
$$\begin{aligned} \begin{pmatrix} \tilde{Q} \\ \tilde{N} \end{pmatrix}= \begin{pmatrix} Q^{(0)} \\ N^{(0)} \end{pmatrix}+ \tilde{K} K_0^{-1}\begin{pmatrix} Q^{(1)} \\ N^{(1)} \end{pmatrix}. \end{aligned}$$
(98)
Notice that we are using the notation \(K_0\equiv K^{(0)}\), and that Eq. (98) is a generalization of Eq. (84). In the particular case \(a^2=b\) we have \(M^{(0)}=\tilde{M}=T^{(0)}=\tilde{T} = 0\) and we then recover the previous definitions of the unitarized \(\tilde{A}\) and \(\tilde{Q}\). In the general case, the amplitudes obtained from Eq. (98) feature all the good properties mentioned above as analyticity in the whole complex plane, left and right cuts, the possibility for developing poles in the second Riemann sheet, etc.