Unitarity in composite Higgs boson approaches with vector resonances

We examine a simple composite Higgs model (CHM) with vector resonances in addition to the Standard Model (SM) fields in perturbation theory by using the $K$-matrix method to implement unitarity constraints. We find that the $W_L{W}_L$ scattering amplitude has an additional scalar pole (analogous to the $\sigma$ meson of QCD) as in generic strongly interacting extensions of the SM. The mass and width of this dynamically generated scalar resonance are large and the mass behaves contrary to the vector one, so that when the vector resonance is lighter, the scalar one is heavier, and vice versa. We also attempt an interpretation of this new resonance. Altogether, the presence of the vector state with the symmetries of the CHM improve the low-energy unitarity behavior also in the scalar-isoscalar channel.

Figure 1

Feynman diagrams that produce the tree-level amplitudes for pion ($W_L$) scattering in the energy range $2m_{W,h}\ll \sqrt{s}\ll 4\pi f\sim$ several TeV.

Figure 2

Modulus of the scalar-isoscalar partial wave $a_0^0$ (a real, positive number for real $m_{\rho }$ and $s$) in Eq. (34) as a function of Mandelstam-$s$. Solid line: reference values of the parameters, $m_{\rho }=2\mathrm{TeV}$, $f=350\mathrm{GeV}$, with the thickness of the line representing the uncertainty in the width ${\mathrm{\Gamma }}_{\rho }\in (5-20\%)$. Thick dashed and dotted lines: same with $m_{\rho }=4$ and 6 TeV. Red thin lines: instead, increase $f$ to 0.7 and 1 TeV (at the larger ${\mathrm{\Gamma }}_{\rho }=0.2m_{\rho }$ width value and $m_{\rho }=2\mathrm{TeV}$).

Figure 3

We plot the scalar-isoscalar partial wave in Eq. (34) explicitly showing the real and imaginary parts. The later comes from the substitution $m_{\rho }\to (m_{\rho }-i\frac{{\mathrm{\Gamma }}_{\rho }}{2})$ with $m_{\rho }=2\mathrm{TeV}$, ${\mathrm{\Gamma }}_{\rho }=20\%m_{\rho }$, $f=350\mathrm{GeV}$.

Figure 4

Modulus of the vector-isovector partial wave $|a_1^1|$ in Eq. (35) as a function of Mandelstam-$s$. Solid lines: ${\mathrm{\Gamma }}_{\rho }=20\%m_{\rho }$. Dotted lines: ${\mathrm{\Gamma }}_{\rho }=5\%m_{\rho }$ for $m_{\rho }=2\mathrm{TeV}$.

Figure 5

Tensor-isoscalar partial wave $a_2^0$. Lines as in Fig. 2.

Figure 6

Scenario 2 converges towards scenario 1 for a certain $g_s$. With the parameters here ($m_{\rho }=2\mathrm{TeV}$, ${\mathrm{\Gamma }}_{\rho }=20\%m_{\rho }$), this happens for $g_s$ slightly larger than 8 if $f=0.35\mathrm{TeV}$ (top plot) and for $g_s$ a bit above 4 for $f=0.7\mathrm{TeV}$ (bottom).

Figure 7

Scenario 2, the complete 4DCHM. Top: imaginary part of the scalar, isoscalar partial wave. Bottom: modulus of the vector, isovector one. Here, $f=350\mathrm{GeV}$ and ${\mathrm{\Gamma }}_{\rho }/m_{\rho }=20\%$.

Figure 8

Test of unitarity for $\mathrm{Im}\tilde{a}=\mathrm{Im}{\tilde{k}}_{11}$ (black solid line). Bottom line (downward triangles, blue online): $|\tilde{m}{|}^2=|{\tilde{k}}_{12}{|}^2$. Second from bottom (upward triangles, red online): $|\tilde{a}{|}^2=|{\tilde{k}}_{11}{|}^2$. Circles (on top of the imaginary part): the sum of the last two, unitarity compliant. Here ${\mathrm{\Gamma }}_{\rho }/m_{\rho }=5\%$, for larger values a small difference is visible due to the perturbative $a_0^0$ acquiring an imaginary part.

Figure 9

Modulus of the unitarized elastic matrix element $|{\tilde{k}}_{11}|=|{\tilde{a}}_0^0(s)|$, that remains below 1 in all the energy interval of interest. First plot (scenario 1): solid lines correspond to $m_{\rho }=2\mathrm{TeV}$, dashed ones to $m_{\rho }=4\mathrm{TeV}$. From thicker to thinner, $f=0.35$, $0.7\mathrm{TeV}$ respectively. Lower plot: in scenario 2, we fix $f=0.35\mathrm{TeV}$ and ${\mathrm{\Gamma }}_{\rho }/m_{\rho }=5\%$. The thick lines (blue online) correspond to $m_{\rho }=2\mathrm{TeV}$, with $g_s=4$ (solid), 6 (dash-dotted) and 8 (dashed). The thin ones (red online) were in turn calculated with $g_s=6$ (solid), 8 (dash-dotted) and 10 (dashed), and all have $m_{\rho }=4\mathrm{TeV}$.

Figure 10

Same as Fig. 9 but for $|{\tilde{k}}_{22}|=|{\tilde{t}}_0^0(s)|$ the elastic $hh\to hh$ scattering that vanishes in the perturbative amplitude at LO. Unitarization requires this channel to have finite probability too.

Figure 11

Top and middle plots: evolution of the mass and width of the dynamically generated scalar $\sigma$-like resonance as function of $f$ in the limit $f_2\to \infty$ (scenario 1), for three values of the vector mass. Bottom: width of the scalar resonance against its mass, with $f$ being now just the parameter of the trajectory in this plane.

Figure 12

Varying $g_s$ produces a curve in the $(m_0,{\mathrm{\Gamma }}_0)$ plane of the $\sigma$-like pole as extracted from its complex-$s$ position, for $f=0.7\mathrm{TeV}$ (top) and $f=0.35\mathrm{TeV}$ (bottom), and for $m_{\rho }=2$ or 4 TeV as indicated. It may happen that for each value of $m_0$ there are two values of ${\mathrm{\Gamma }}_0$. This reflects the oscillation with the parameter of the curve, $g_s$, visible in Fig. 13, and caused by the nontrivial dependence of Eq. (31) on $g_s$.

Figure 13

Evolution of the mass (top) and width (bottom) of the dynamically generated scalar $\sigma$-like resonance as a function of $g_s$ in scenario 2, for fixed vector masses 2 and 4 TeV and fixed $f=0.35$ and 0.7 TeV as indicated.

Figure 14

Scalar-isoscalar perturbative amplitude of $\pi \pi \to {\rho }_L{\rho }_L$ for $m_{\rho }=2\mathrm{TeV}$, with $g_s=3$ (dashed line) and $g_s=4$ (solid line) in scenario 2. The two bottom lines and the two top lines correspond respectively to $f=0.35$ and 0.7 TeV.