Electroweak restoration at the LHC and beyond: The $Vh$ channel

The LHC is exploring electroweak (EW) physics at the scale EW symmetry is broken. As the LHC and new high energy colliders push our understanding of the Standard Model to ever-higher energies, it will be possible to probe not only the breaking of but also the restoration of EW symmetry. We propose to observe EW restoration in double EW boson production via the convergence of the Goldstone boson equivalence theorem. This convergence is most easily measured in the vector boson plus Higgs production, $Vh$, which is dominated by the longitudinal polarizations. We define EW restoration by carefully taking the limit of zero Higgs vacuum expectation value (vev). EW restoration is then measured through the ratio of the $p_T^h$ distributions between $Vh$ production in the Standard Model and Goldstone boson plus Higgs production in the zero vev theory, where $p_T^h$ is the Higgs transverse momentum. As EW symmetry is restored, this ratio converges to one at high energy. We present a method to extract this ratio from collider data. With a full signal and background analysis, we demonstrate that the 14 TeV HL-LHC can confirm that this ratio converges to one to 40% precision while at the 27 TeV HE-LHC the precision will be 6%. We also investigate statistical tests to quantify the convergence at high energies. Our analysis provides a roadmap for how to stress test the Goldstone boson equivalence theorem and our understanding of spontaneously broken symmetries, in addition to confirming the restoration of EW symmetry.

Figure 1

Ratio of transverse momentum distributions of polarized gauge boson production to the total distribution summed over polarizations. These are shown using $W$ transverse momentum, $p_T^W$, for (a) $W^{\pm }{W}^{\mp }$ and (b) $W^{\pm }Z$; and Higgs transverse momentum, $p_T^h$, for (c) $W^{\pm }h$ and (d) $Zh$. The gauge boson polarizations are (blue dash-dot-dot) fully longitudinal, (black solid) fully transverse, and (red dashed) $\mathrm{longitudinal}+\mathrm{transverse}$. The lab frame energy is the HL-LHC energy of $\sqrt{S}=14\mathrm{TeV}$. The subscript $T$ on the gauge bosons indicate summed over transverse polarizations.

Figure 2

Signal strengths (black solid) ${\mu }_{Wh}$, (red dashed) ${\mu }_{Zh}$ at $\sqrt{S}=14\mathrm{TeV}$ and (magenta dotted) ${\mu }_{Wh}$, (blue dash-dot-dot) ${\mu }_{Zh}$ at $\sqrt{S}=27\mathrm{TeV}$.

Figure 3

The number of events for (red line) signal and (colored bars) background at 27 TeV with $15{\mathrm{ab}}^{-1}$ of data. The background is cumulative and the signal overlaid. We show (a,b) two lepton, (c,d) one lepton, and (e,f) zero lepton signal categories for (left-hand side) two jet and (right-hand side) three jet channels. The last bin is an overflow bin. $V+jj$ backgrounds include $V+HF$, $V+cl$, and $V+ll$. Here, $ZV$ and $WV$ (Top) indicate $V{V}^{\prime }$ ($tt$ and single top) backgrounds with no missing leptons. “Missing lepton” indicates backgrounds where a lepton is missed, which is all other backgrounds except those explicitly listed.

Figure 4

Signal strengths ${\mu }_{Vh}$ central values and 68% CL for (a,b) two lepton categories, (c,d) one lepton categories, and (e,f) zero lepton categories for (a,c,e) 14 TeV with $3{\mathrm{ab}}^{-1}$ and (b,d,f) 27 TeV with $15{\mathrm{ab}}^{-1}$. We show the (red) two jet and (gray) three jet categories separately.

Figure 5

Combined central values across all categories and 68% CL for extracted signal strengths ${\mu }_{Vh}$ at (a) 14 TeV with $3{\mathrm{ab}}^{-1}$ and (b) 27 TeV with $15{\mathrm{ab}}^{-1}$. Black dashed lines are the partonic level prediction, the red bars are statistical uncertainty, the green bars are statistical and a 5% systematic uncertainty added in quadrature.

Figure 6

The ${\chi }^2$ per degree of freedom $\mathrm{\Delta }{\chi }_m^2$ defined in Eq. (24) (left panel) KL divergence defined in Eq. (27) (right panel) for 14 TeV with $3{\mathrm{ab}}^{-1}$ and 27 TeV with $15{\mathrm{ab}}^{-1}$. The back (red dashed) line represents the median values of the 27 TeV (14 TeV) results, and the yellow and green bands represent the values where 68% and 95% of pseudoexperiments lie, respectively.