Proposals of measuring the off-shell Higgs contributions and first measurements at the LHC have electrified the Higgs phenomenology community for two reasons: firstly, probing interactions at high invariant masses and momentum transfers is intrinsically sensitive to new physics beyond the Standard Model, irrespective of a resonant or non-resonant character of a particular BSM scenario. Secondly, under specific assumptions a class of models exists for which the off-shell coupling measurement together with a measurement of the on-shell signal strength can be re-interpreted in terms of a bound on the total Higgs boson width. In this paper, we provide a first step towards a classification of the models for which a total width measurement is viable and we discuss examples of BSM models for which the off-shell coupling measurement can be important in either constraining or even discovering new physics in the upcoming LHC runs. Specifically, we discuss the quantitative impact of the presence of dimension six operators on the (de)correlation of Higgs on- and off-shell regions keeping track of all interference effects. We furthermore investigate off-shell measurements in a wider context of new (non-)resonant physics in Higgs portal scenarios and the MSSM.
Article funded by SCOAP3
m_h$ (section~\ref{sec:res}).]]>
\Gamma_h^{\text{SM}}\simeq 4~\text{MeV}$, yet still a SM value for the $pp\to h\to ZZ$ signal strength $\mu^{\text{on}}_{ZZ}$, we need to have $\kappa^2_{ggh}\, \kappa^2_{hZZ}>1$. If we consider an extrapolation of the on-shell region to the off-shell region based on the SM Feynman graph templates depicted in figure~\ref{fig:feyn}, we can understand a constraint on $\overline\sigma_h $ as a constraint on $\Gamma_h$ {\emph{as a consistency check}}: in a well-defined QFT framework such as the SM, a particle width is a consequence of the interactions and degrees of freedom as specified in the Lagrangian density. E.g. by extending the SM with dynamics that induce an invisible partial Higgs decay width, there is no additional information in the off-shell measurement when combined with the on-shell signal strength. It is important to note that if we observe an excess in $\overline\sigma_h$ in the future, then this will not be a manifestation of $\Gamma_h>\Gamma_h^{\text{SM}}$. Instead we will necessarily have to understand this as a observation of physics beyond the SM, which might but does not need to be in relation to the Higgs boson. A quantitatively correct estimate of important interference effects that shape $\overline \sigma_h$ have been provided in refs.~\cite{Kauer-ml-2013qba,Kauer-ml-2013qba.m001,Kauer-ml-2013qba.m002,ciaran,ciaran.m001,ciaran.m002,mit} (see also~\cite{Dixon-ml-2013haa2,Dixon-ml-2013haa2.m001,Dixon-ml-2013haa2.m002,Dixon-ml-2013haa2.m003} for a related discussion of $pp\to h\to \gamma\gamma$). These interference effects are an immediate consequence of a well-behaved electroweak sector in the sub-TeV range in terms of renormalizability and, hence, unitarity~\cite{bsm,Chanowitz-ml-1978mv,Chanowitz-ml-1978mv.m001}. While they remain calculable in electroweak leading order Monte Carlo programs~\cite{Kauer-ml-2013qba,Kauer-ml-2013qba.m001,Kauer-ml-2013qba.m002,ciaran,ciaran.m001,ciaran.m002}, they are not theoretically well-defined, unless we assume a specific BSM scenario or invoke EFT methods. For a discussion on the unitarity constraints on the different Wilson coefficients see~\cite{kramer}. Both ATLAS and CMS have performed the outlined measurement with the 8\,TeV data set in the meantime~\cite{cmswidth,atlaswidth}. The importance of high invariant mass measurements in this particular channel in a wider context has been discussed in~refs.~\cite{fermilab1,fermilab1.m001,fermilab1.m002,fermilab2,bsm,bsmh,bsmh.m001,bsmh2,bsmh2.m001} In the particular case of $pp\to ZZ\to 4\ell$, we can classify models according to their effect in the on-shell and off-shell phase space regions. We can identify four regions depending on the measured value of $\mu^{\text{off}}_{ZZ}$, which can provide a strong hint for new physics in the above scenarios (ii)--(iv): \begin{equation} \label{eq:interp} \parbox{8cm}{\vspace{-0.4cm} \begin{enumerate} \item \qquad $\mu^{\text{off}}_{ZZ} = 1$ and $[\kappa^2_{ggh}\kappa^2_{hZZ}]^{\text{on}} = 1$\,, \item \qquad $\mu^{\text{off}}_{ZZ} = 1$ and $[\kappa^2_{ggh}\kappa^2_{hZZ}]^{\text{on}} \neq 1$\,, \item \qquad $\mu^{\text{off}}_{ZZ} \neq 1$ and $[\kappa^2_{ggh}\kappa^2_{hZZ}]^{\text{on}} = 1$\,, \item \qquad $\mu^{\text{off}}_{ZZ} \neq 1$ and $[\kappa^2_{ggh}\kappa^2_{hZZ}]^{\text{on}} \neq 1$\,. \end{enumerate} \vspace{-0.4cm} } \end{equation} We can write a generalized version of eq.~(\ref{eq:1b}) that also reflects (non-)resonant BSM effects by writing the general amplitude \begin{multline} \label{eq:ggzz} {\cal{M}}(gg\to ZZ) = \bigg[ [g_{hZZ} g_{ggh}](\hat s, \hat t) + [\tilde g_{hZZ} \tilde g_{ggh} ](\hat s, \hat t) +\sum_i [\tilde g_{ggX_i} \tilde g_{X_iZZ}](\hat s, \hat t)\bigg]\\ + \big\{ g_{ggZZ}(\hat s, \hat t) + \tilde g_{ggZZ} (\hat s, \hat t) \big\}\,, \end{multline} from which we may compute $\hbox{d} \overline \sigma(gg)\sim |{\cal{M}}|^2$ by folding with parton distribution functions and the phase space weight. For $\bar q q$-induced $ZZ$ production we can formulate a similar amplitude \begin{equation} {\cal{M}}(\bar qq\to ZZ) = g_{\bar q q ZZ} (\hat s, \hat t) + \tilde g_{\bar q q ZZ} (\hat s, \hat t) + \sum_i \tilde [g_{\bar q q X_i} \tilde g_{X_iZZ}](\hat s, \hat t)\,, \end{equation} which can impact the $Z$ boson pair phenomenology on top of the $gg$-induced channels. Hence, for the differential off-shell cross section we find $\hbox{d} \overline \sigma \simeq \hbox{d} \overline \sigma(gg) + \hbox{d} \overline \sigma(\bar q q)$. Resonant scenarios, such as new scalars and vectors are in agreement with the generalized Landau-Yang theorem~\cite{landau} have been studied in detail~\cite{more,more.m001,more.m002,more.m003}. Non-resonant new interactions involving light quarks, e.g.\ in a dimension six operator extension of the SM, are typically constrained. For all models that fall into the classification 1. we are allowed to re-interpret the off-shell measurement as a constraint on the Higgs width bearing in mind theoretical shortcomings when parameters are varied inconsistently; the uncertainty of a measurement of $\mu^{\text{off}}_{ZZ}$ and the on-shell signal strength $\mu^{\text{on}}_{ZZ}$ combine to a constraint on $\Gamma_h$. Assuming new physics exists, such a constraint makes strong assumptions about potential cancellations among or absence of the new physics couplings in the off-shell region. In particular because the effective couplings are phase space dependent and can affect the differential $m_{ZZ}$ distribution beyond a simple rescaling. A concrete example of this class of models is the general dimension six extension of the SM Higgs sector with a Higgs portal to provide an invisible partial decay width $\Gamma^{\text{inv}}$. If we are in the limit of vanishing dimension six Wilson coefficients $c_i\ll v^2/f^2$, new EFT physics contributions with new physics scale $f$ in the on- and off-shell regions are parametrically suppressed and the dominant unconstrained direction in this measurement is $\Gamma^{\text{inv}}$. Note that there can be cancellations in the high invariant mass region among different dimension six coefficients, so the constraint formulated on $\Gamma^{\text{inv}}$ \emph{requires} $c_i\to 0$. For the second scenario a re-interpretation in terms of a width measurement is generally not valid. Here, the SM off-shell distribution is recovered while the on-shell signal strength is unity due to a cancellation between the modified Higgs width and the on-shell coupling modification. A toy-model example has been discussed in~\cite{bsm}. From a phenomenological point of view, scenarios 3. and 4. are of great interest, in particular because SM-like signal strength measurements alone do typically not provide enough information to rule out models conclusively. Most concrete realizations of BSM physics predict new physics at high energies as a unitarity-related compensator for modifications of on-shell coupling strengths. ``Off-shell'' measurements are therefore prime candidates to look for deviations from the Standard Model in the sense that they will be sensitive to new resonances~\cite{bsmres,bsmres.m001,bsmres.m002,bsmres.m003} and will have strong implications for BSM physics in~general. The aim of this work is to provide a survey of the reach of the validity of the Higgs width interpretation. Since modifications of the Higgs width do imply physics beyond the SM, the Higgs width interpretation can be reconciled with new physics effects in the $ZZ$ channel. This allows us to make contact to concrete phenomenological realizations using the above categorization. New degrees of freedom as introduced in the beginning of this section that give rise to new contributions following eq.~\eqref{eq:ggzz}. We focus on $gg$-induced $ZZ$ production throughout. We will first discuss light non-resonant degrees of freedom and their potential impact on the $m_{ZZ}$ distributions with the help of toy models that we generalize to the (N)MSSM in section~\ref{sec:ldof}. Assuming a scale separation between new resonant phenomena and the probed energy scales in $pp\to ZZ \to 4\ell$ we discuss high invariant mass $Z$ boson pair production in a general dimension six extension of the SM in section~\ref{sec:eft} before we consider resonant phenomena in section~\ref{sec:res}. In particular, our calculation includes all interference effects (at leading order) of $pp\to ZZ\to 4\ell$ in all of these scenarios. Our discussions and findings straightforwardly apply to the $WW$ channel which is, due to custodial symmetry, closely related to the $ZZ$ final state. ]]>
m_h$. As demonstrated in figure~\ref{fig:susy}, the effects quickly decouple with larger stop masses and smaller values of $A_t\lesssim 1~\text{TeV}$. ]]>
330~\text{GeV})$. If we invoke the operator coefficient bias and of eq.~\eqref{lsilh} focus on a tree-level $T$ parameter $T=0$, the dominant operator coefficients that are probed in the off-shell region are $c_H,c_g,c_t$. A targeted analysis of how far these parameters can be constrained at the LHC has been presented in refs.~\cite{bsmh2,bsmh2.m001}; a question that remains worth addressing in this context, however, is the impact of the off-shell measurement in comparison to Higgs measurements in other channels such as associated Higgs~\cite{Butterworth-ml-2008iy,asso} and Higgs+jet~\cite{monojet1,monojet1.m001,monojet1.m002} production. In the following we input the SM-like signal strengths in the $pp\to ZZ$ channel since direct measurements in the latter channels are not available at 8\,TeV. The signal distributions for a representative operator choice $c_g\simeq 0.25 v^2/\Lambda^2$ is given in figure~\ref{fig:eftcomp} and we use eHdecay~\cite{dim6gg1} to compute the modified branching ratios, inputing the the bigger Higgs width to achieve $\mu^{\rm on}_{ZZ}=1$. The different thresholds and normalizations in figure~\ref{fig:eftcomp} reflect the signal regions and selection efficiencies as documented in the literature~\cite{asso,monojet,monojet.m001,monojet.m002} due to $b$-tagging, $\tau$ reconstruction and subjet techniques. It should be noted that associated Higgs and Higgs+jet production are plagued with large backgrounds as opposed to the experimentally clean $ZZ\to 4\ell$ signature,\footnote{For instance, a measurement of the off-shell cross section is already available with the 8\,TeV data set although the inclusive signal cross section is significantly smaller compared to $Z$-associated and jet-associated Higgs production.} the signal-to-background ratio in e.g.\ $pp\to hj\to \tau^+\tau^-$ is of the order of 0.1~\cite{monojet,monojet.m001,monojet.m002}. A measurement of the differential distributions as shown in figure~\ref{fig:eftcomp} in these channels will be complicated: while the acceptance in the fully leptonic $ZZ$ final state at large invariant four-lepton masses is close to unity~\cite{atlaswidth,cmswidth}, the signal rates in associated and monojet production are vastly reduced (for details see e.g.~\cite{asso} and~\cite{monojet,monojet.m001,monojet.m002}). Therefore, off-shell measurements in the $pp\to ZZ$ channel will not only provide crucial information to limit the presence of higher dimensional operators but also provide complementary information, in particular due to a larger kinematically accessible phase space range. \begin{figure} \centerline{ \includegraphics[width=0.45\textwidth]{plot_mzz_13_cmp.eps}} ]]>
m_h &:\quad \text{coupling suppression $\sin\chi$} \end{alignat} and keep the Higgs width identical to the SM (this could be facilitated by another portal interaction to light SM-singlet states). This will modify the on-shell Higgs phenomenology and we choose $\mu^{\text{on}}_{ZZ}=\cos^4\chi=0.81$, which is within the $H\to ZZ$ limits as reported in latest coupling fits in the $ZZ$ category~(see e.g.~\cite{hzz,hzz.m001,hzz.m002}). This choice is also consistent with the non-observation of a heavy Higgs-like particle with a signal strength of $\sim 10\%$ of the SM expectation in a region where the narrow width approximation is valid (see e.g.\ recent searches by CMS~\cite{Chatrchyan-ml-2013yoa}) and limits set by electroweak precision constraints; see also~\cite{bsmres2} for a detailed discussion of currently allowed parameter range, and~\cite{portal0} on constraints that can be obtained by measuring the heavy Higgs boson. Since the light Higgs width quickly decouples this choice is irrelevant for the phenomenology at high invariant mass. To keep our discussion transparent, we choose a trivial hidden sector phenomenology by using \begin{equation} \Gamma_\phi(m_\phi)= \sin^2\chi \, \Gamma_h^{\text{SM}}(m_\phi) \end{equation} in the following. The results for two representative choices of $m_\phi$ are shown in figure~\ref{fig:wq}. The structure in the ``$H+\phi$'' signal results from a destructive interference of the Higgs diagrams in the intermediate region $m_h < \sqrt{\sh} \lesssim m_\phi$ as a consequence of the propagator structure and will depend on how we formulate the Higgs width theoretically~\cite{Hwidth,Hwidth.m001}.\footnote{A survey of dip structures in cross sections has been presented in refs.~\cite{Bai-ml-2014fkl,Stuart-ml-1991xk,Stuart-ml-1991xk.m001,Stuart-ml-1991xk.m002}.} From a phenomenological perspective this structure is numerically irrelevant. Apart from the obvious additional resonance, we do not find a notable deviation from the SM away from the Breit-Wigner ``turn on'' region $m(4\ell)\gtrsim m_\phi$. Away from all $s$-channel particle thresholds, i.e.\ for invariant masses $m(4\ell)\gg m_\phi$, the amplitude becomes highly resemblant to the SM amplitude as a consequence of the linear mixing: if we write the SM top-triangle subamplitude as ${\cal{C}}(\hat s,m^2_t)$ and remove the $Z$ boson polarization vectors, we have an amplitude \begin{multline} \label{eq:cos} {\cal M}^{\mu\nu} = g^{\mu\nu} {\cal{C}}(\hat s,m_t^2) \times\left( {\cos^2\chi\over \sh - m_h^2 +i m_h\Gamma_h} + {\sin^2\chi\over \sh - m_\phi^2 +i m_\phi\Gamma_\phi} \right)\\ \to {g^{\mu\nu}\over \sh}{\cal{C}} (\hat s,m_t^2)\quad\hbox{for $\sh\gg m_h^2,m_\phi^2$,} \end{multline} which is just the SM contribution evaluated at large $\sqrt{\hat s}$. This qualitative argument is numerically validated for the full cross section in figure~\ref{fig:wq2}. The differential $m_{ZZ}$ distribution approaches the SM distribution rather quickly, especially because consistency with the 125\,GeV signal strength measurements and electroweak precision data~\cite{elwp} imposes a hierarchy $\cos^2\chi \gg \sin^2\chi$. Eq.~\eqref{eq:cos} suggests that the more interesting parameter choice for modified interference effects at large invariant masses is a larger mixing. In this case, however, the Higgs on-shell phenomenology would vastly modified too. Larger values of $\sin^2\chi$ also imply tension with electroweak precision data and direct search constraints, unless we give up the simplified model of eq.~\eqref{eq:portal}. This is beyond the scope of this work. Quantitatively a larger mixing only shows a moderate increase for $m(4\ell)\gtrsim 400$\,GeV (we include a maximum mixing angle $\cos^2\chi=0.5, m_\phi=350~\text{GeV}$ to figure~\ref{fig:wq2}), which results from Breit-Wigner distribution of the state $\phi$; for maximal mixing this has a larger signal strength compared to the $\cos^2\chi=0.9$~scenario. In summary, we conclude that the basic arguments that have been used in the interpretation of SM measurements~\cite{Kauer-ml-2013qba,Kauer-ml-2013qba.m001,Kauer-ml-2013qba.m002,ciaran,ciaran.m001,ciaran.m002,mit,Dixon-ml-2013haa2,Dixon-ml-2013haa2.m001,Dixon-ml-2013haa2.m002,Dixon-ml-2013haa2.m003,cmswidth,atlaswidth} remain valid in this minimal resonant extension of the SM Higgs sector. Our analysis straightforwardly generalizes to the two Higgs doublet model~\cite{bib2hdm} and the $n$HDM~\cite{nhdm}. ]]>