Precise predictions for the Higgs-boson decay H -> W W/Z Z -> 4 leptons

TL;DR

This paper delivers a comprehensive calculation of electroweak radiative corrections to SM Higgs decays into four leptons via off-shell $W$ and $Z$ bosons, incorporating full ${\cal O}(\alpha)$ effects plus heavy-Higgs and final-state radiation improvements. The complex-mass scheme is employed to describe intermediate gauge-boson resonances without expanding around on-shell propagators, and the results are implemented in the PROPHECY4f Monte Carlo generator. Corrections to partial widths are typically a few percent and grow with the Higgs mass, reaching about 8% near $M_H \sim 500$ GeV, with larger distortions in angular and invariant-mass distributions due to radiation effects. An Improved-Born Approximation (IBA) and NWA comparisons are provided to gauge the size of higher-order effects, showing good agreement (within ~2%) for $M_H \lesssim 400$ GeV, while the full off-shell treatment remains essential near thresholds. These precise predictions are important for Higgs characterization at the LHC and future colliders, enabling accurate determinations of branching fractions, angular correlations, and resonance structures in $H\to 4f$ decays.

Abstract

The decay of the Standard Model Higgs boson into four leptons via a virtual W-boson or Z-boson pair is one of the most important decay modes in the Higgs-boson search at the LHC. We present the complete electroweak radiative corrections of O(α) to these processes, including improvements beyond O(α) originating from heavy-Higgs effects and final-state radiation. The intermediate W- and Z-boson resonances are described (without any expansion or on-shell approximation) by consistently employing complex mass parameters for the gauge bosons (complex-mass scheme). The corrections to partial decay widths typically amount to some per cent and increase with growing Higgs mass M_H, reaching about 8% at M_H \sim 500 GeV. For not too large Higgs masses (M_H <\sim 400 GeV) the corrections to the partial decay widths can be reproduced within <\sim 2% by simple approximations. For angular distributions the corrections are somewhat larger and distort the shapes. For invariant-mass distributions of fermion pairs they can reach several tens of per cent depending on the treatment of photon radiation. The discussed corrections have been implemented in a Monte Carlo event generator called PROPHECY4F.

Figures (4)

• Figure 1: Generic lowest-order diagram for ${\rm H}$ H$\to 4f$ where $V={\rm W}$ W$,{\rm Z}$ Z$$.
• Figure 2: Dependence of the relative corrections $\delta$ to the partial decay width on the energy cutoff $\Delta E$ (l.h.s.) and on the angular cutoff $\Delta\theta$ (r.h.s.) in the slicing approach for the decay ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+$\mu^-\bar{\nu}_\mu$ with $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$. For comparison the corresponding result obtained with the dipole subtraction method is shown as a $1\sigma$ band in the plots.
• Figure 3: Same as in Figure \ref{['fig:sliWW']} but for the decay ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+$\mu^-\mu^+$.
• Figure 4: Partial decay width for ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ as a function of the Higgs mass. The upper plots show the absolute prediction including $\cal{O}\alpha)$O(α)$$ and ${\cal O}(G\mu$G_μ$^2M{\rm H}$ H$$M_${\rm H}$$^4)$ corrections, and the lower plots show the comparison of the corresponding relative corrections with the NWA and IBA. The lower plots show the corrections relative to the lowest-order result. As already explained, we always normalize to the lowest-order result that already includes the $\cal{O}\alpha)$O(α)$$-corrected gauge-boson width in the complex masses of the gauge bosons. A large fraction of the $\cal{O}\alpha)$O(α)$$ corrections is transfered to the lowest-order decay width by applying the $G\mu$G_μ$$ scheme. Thus, the corrections are at the order of 2--8% for moderate Higgs masses. However, for large Higgs masses the corrections become larger and reach about 13% at $M{\rm H}$ H$$M_${\rm H}$$=700\,{\rm GeV}$. In this region the leading two-loop corrections already amount to about 4%. Around the ${\rm W}$ W${\rm W}$ W$$ threshold at $160\,{\rm GeV}$ the Coulomb singularity, which originates from soft-photon exchange between the two slowly moving W bosons, is reflected in the shape of the curve. The influence of diagrams with a Higgs boson splitting into a virtual Z-boson pair (ZZ threshold) is visible at $M{\rm H}$ H$$M_${\rm H}$$\sim2M{\rm Z}$ Z$$M_${\rm Z}$$$. At about $2m{\rm t}$ t$$m_${\rm t}$$$ the ${\rm t}$ t${\rm \bar{t}}$ t̅$$ threshold is visible. For stable W or Z bosons, i.e. in the limit $\Gamma_V\to 0$ ($V={\rm W}$ W$,{\rm Z}$ Z$$), it is possible to define a narrow-width approximation (NWA) where the matrix elements factorize into the decay ${\rm H}$ H$\to VV$ and the subsequent decay of the gauge bosons into fermions. By definition the NWA is only applicable above the WW or ZZ threshold. However, its analytical structure and evaluation is considerably simpler than in the case of the full decay ${\rm H}$ H$\to{\rm W}$ W${\rm W}$ W$/{\rm Z}$ Z${\rm Z}$ Z$\to4f$ with off-shell gauge bosons. Therefore, above threshold the NWA allows for an economic way of calculating relative $\cal{O}\alpha)$O(α)$$ corrections to the integrated decay width, while the lowest-order contribution may, of course, still take into account unstable gauge bosons. Following this line of thought, we define $\Gamma^{{\rm }{\rm NWA}} = \Gamma_0 \, \frac{\Gamma^{{\rm }{\rm NWA}}_1}{\Gamma^{{\rm }{\rm NWA}}_0},$with $\Gamma^{{\rm }{\rm NWA}}_1 = \Gamma_{{\rm H}{\rm H} VV,1}\, \frac{\Gamma_{V f_1\bar{f}_2,1} \Gamma_{V f_3\bar{f}_4,1}}{\Gamma_{V,1}\Gamma_{V,1}},$and $\Gamma^{{\rm }{\rm NWA}}_0 = \Gamma_{{\rm H}{\rm H} VV,0}\, \frac{\Gamma_{V f_1\bar{f}_2,0} \Gamma_{V f_3\bar{f}_4,0}}{\Gamma_{V,1}\Gamma_{V,1}}.$The indices "0" and "1" label lowest-order and $\cal{O}\alpha)$O(α)$$-corrected results, respectively. The Higgs-mass-enhanced two-loop terms, described in Section \ref{['se:gf2mh4']}, have also been included in $\Gamma_{{\rm H}$ H$VV,1}$. In order to be consistent we again use the $\cal{O}\alpha)$O(α)$$-corrected total width for the gauge bosons in the denominators of the branching ratios in $\Gamma^{{\rm }$ NWA$}_0$. We note that we have rederived all necessary $\cal{O}\alpha)$O(α)$$ corrections entering the NWA; the hard-photon corrections to the decay ${\rm H}$ H$\to{\rm W}$ W${\rm W}$ W$$ have been successfully checked against the expression given in Ref. Kniehl:1991xe. The NWA is evaluated with real gauge-boson and top-quark masses. A few GeV above the corresponding gauge-boson-pair threshold the NWA agrees with the complete corrections within 1%. Near $M{\rm H}$ H$$M_${\rm H}$$=2M{\rm Z}$ Z$$M_${\rm Z}$$\sim180\,{\rm GeV}$ the loop-induced ZZ threshold can be seen in the relative corrections to ${\rm H}$ H$\to{\rm W}$ W${\rm W}$ W$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ shown in Figure \ref{['fig:sqrtsww']}. In the NWA this threshold leads to a singularity visible as a sharp peak; in the off-shell calculation in the complex-mass scheme this singular structure is smeared out, because the finite Z-boson width is taken into account. Since the ZZ threshold corresponds to the situation where two Z bosons become on shell in the loop, the latter description with the singularity regularized by a finite $\Gamma_{{\rm Z}$ Z$}$ is closer to physical reality. An analogous situation can be seen near $M{\rm H}$ H$$M_${\rm H}$$=2m{\rm t}$ t$$m_${\rm t}$$\sim350\,{\rm GeV}$ for the ${\rm t}$ t$\bar{{\rm t}$ t$}$ threshold with top quarks in the loops. Again the inclusion of the top decay width $\Gamma_{{\rm t}$ t$}$, as done in the off-shell calculation, yields the better description. In Figure \ref{['fig:sqrtsww']} we show also the relative corrections obtained in the IBA of (\ref{['eq:IBA']}). The Coulomb singularity and the ${\rm t}$ t$\bar{{\rm t}$ t$}$ threshold are well described, and the full corrections are reproduced within $\mathrel{\raisebox{-.3em}{$\stackrel{ <}{\sim}$}}2\%$ for Higgs masses below $400\,{\rm GeV}$. The plots in Figure \ref{['fig:sqrtszz']} show the decay width and the relative correction for the final state ${\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$. Partial decay width for ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ as a function of the Higgs mass. The upper plots show the absolute prediction including $\cal{O}\alpha)$O(α)$$ and ${\cal O}(G\mu$G_μ$^2M{\rm H}$ H$$M_${\rm H}$$^4)$ corrections, and the lower plots show the comparison of the corresponding relative corrections with the NWA and IBA.The corrections are between 2% and 4% for moderate Higgs masses and rise to more than 10% for large Higgs masses. At a Higgs mass of about $160\,{\rm GeV}$ the influence of the ${\rm W}$ W${\rm W}$ W$$ threshold can be observed. As explained above, the behaviour of the corrections as a function of the Higgs mass is smooth, because the finite W-boson width is also used in the loop integrals. In contrast to the decay ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$, there is no Coulomb singularity in this channel because the Z boson is electrically neutral. The NWA reproduces the complete result within $0.5\%$ not too close to the threshold. The IBA agrees with the complete calculation to better than 2% for not too large Higgs masses. Predictions for the partial decay widths of the Higgs boson can also be obtained with various program packages, such as HdecayDjouadi:1997yw. Hdecay contains the lowest-order decay width for ${\rm H}$ H$\to V^{(*)}V^{(*)}$ and the leading one-loop corrections $\propto G\mu$G_μ$M{\rm H}$ H$$M_${\rm H}$$^2$ and two-loop corrections $\propto G\mu$G_μ$^2M{\rm H}$ H$$M_${\rm H}$$^4$. In order to obtain the decay width for ${\rm H}$ H$\to{\rm W}$ W${\rm W}$ W$/{\rm Z}$ Z${\rm Z}$ Z$\to 4f$, we define $\Gamma^{{\rm HD}} = \Gamma_{{\rm H}{\rm H} VV}^{{\rm HD}} \, \frac{\Gamma_{V f_1f_2,0}}{\Gamma_{V,1}} \, \frac{\Gamma_{V f_3f_4,0}}{\Gamma_{V,1}},$where $\Gamma_{{\rm H}$ H$VV}^{{\rm HD}}$ is the decay width from Hdecay. In (\ref{['eq:HD']}) the branching ratios of the gauge bosons are normalized in the same way (lowest order in the numerator, corrected total width in the denominator) as the effective branching ratios of our lowest-order predictions for the ${\rm H}$ H$\to VV\to4f$ partial widths; otherwise a comparison would not be conclusive. The comparison in Figure \ref{['fig:sqrts_hdec']}, where $\Gamma^{{\rm HD}}$ is shown relative to our complete lowest-order prediction, shows that Hdecay agrees with our lowest-order prediction below the decay threshold quite well. Predictions for the partial decay widths for ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ and ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ obtained with the program Hdecay normalized to the complete lowest-order decay width. The corrections shown in Figures \ref{['fig:sqrtsww']} and \ref{['fig:sqrtszz']} are included for comparison. In this region $\Gamma_{{\rm H}$ H$VV}^{{\rm HD}}$ consistently takes into account the off-shell effects of the gauge bosons. Above the threshold Hdecay neglects off-shell effects of the gauge bosons. For large $M{\rm H}$ H$$M_${\rm H}$$$ it follows our corrected result within a few per cent, because the dominant radiative corrections $\propto G\mu$G_μ$M{\rm H}$ H$$M_${\rm H}$$^2$ and $\propto G\mu$G_μ$^2M{\rm H}$ H$$M_${\rm H}$$^4$, which grow fast with increasing $M{\rm H}$ H$$M_${\rm H}$$$, are included in both calculations. In the threshold region, off-shell effects are, however, very important. Here, the difference between the complete off-shell result and the Higgs width for on-shell gauge bosons amounts to more than 10%. In detail, Hdecay interpolates between the off-shell and on-shell results within a window of $\pm2\,{\rm GeV}$ around threshold. The maxima in the Hdecay curves near the WW and ZZ thresholds in the upper and lower left plots of Figure \ref{['fig:sqrts_hdec']}, respectively, are artifacts originating from the on-shell phase space of the W or Z bosons above threshold. Approaching the threshold from above, the on-shell phase space, and thus the corresponding partial decay width, tends to zero. This feature is avoided in Hdecay by the interpolation. The described maxima in the Hdecay curves have nothing to do with the maximum of the correction near the WW threshold in the upper left plot, which is due to the Coulomb singularity. In Figure \ref{['fig:winv']} we study the invariant-mass distribution of the fermion pairs resulting from the decay of the W bosons in the decay ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$. Distribution in the invariant mass of the ${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ (l.h.s.) pair and relative corrections to the distributions in the invariant masses of the $\nu$ν$_{{\rm e}}{\rm e^+}$ e^+$$ and ${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ pairs (r.h.s.) in the decay ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ for $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$ and $M{\rm H}$ H$$M_${\rm H}$$=140\,{\rm GeV}$. The plots on the l.h.s. show the distribution for ${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ including corrections for $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$ and $M{\rm H}$ H$$M_${\rm H}$$=140\,{\rm GeV}$, i.e. for one value of $M{\rm H}$ H$$M_${\rm H}$$$ above and one below the WW threshold. The plots on the r.h.s. compare the relative corrections for $\nu$ν$_{{\rm e}}{\rm e^+}$ e^+$$ and ${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ both with and without photon recombination. The invariant mass $M_{f\bar{f}'}$ is calculated from the sum of the momenta of the fermions $f$ and $f'$. If no photon recombination is applied, always the bare momenta are taken. In the case of photon recombination the momentum of recombined photons is included in the invariant mass as described in Section \ref{['se:input']}. For $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$, where the width is dominated by contributions where both intermediate W bosons are simultaneously resonant, the shape of the curves in Figure \ref{['fig:winv']} can be understood as follows. If one of the fermions resulting from the decay of a resonant W boson emits a photon, the invariant mass $M_{f\bar{f}'}$ is reduced, giving rise to an enhancement for small invariant masses. Without photon recombination these positive corrections are large due to the appearance of logarithms of the small fermion masses. As the electron mass is smaller, the corresponding logarithms yield a larger contribution. If photon recombination is applied, events are rearranged from small invariant masses to large invariant masses. In this case, the observable is inclusive, i.e. the fermion mass logarithms cancel owing to the KLN theorem, and the $\nu$ν$_{{\rm e}}{\rm e^+}$ e^+$$ and ${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ distributions do not differ. The analogous phenomenon has, e.g., been discussed for the related resonance processes ${\rm e^+}$ e^+${\rm e^-}$ e^-$\to{\rm W}$ W${\rm W}$ W$\to 4\,$leptons Denner:2000bjBeenakker:1998gr and $\gamma\gamma\to{\rm W}$ W${\rm W}$ W$\to 4f$Bredenstein:2005zk. For $M{\rm H}$ H$$M_${\rm H}$$=140\,{\rm GeV}$, i.e. below the threshold, only one W boson can become on shell. Thus, there is still a resonance around $M_{f\bar{f}'}\sim M{\rm W}$ W$$M_${\rm W}$$$, but also an enhancement below an invariant mass of $M{\rm H}$ H$$M_${\rm H}$$-M{\rm W}$ W$$M_${\rm W}$$\sim60\,{\rm GeV}$, where the other decaying W boson can become resonant. Near the resonance at $M_{f\bar{f}'}\sim M{\rm W}$ W$$M_${\rm W}$$$ the corrections look similar to the doubly-resonant case discussed for $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$ above. The same redistribution of events from higher to lower invariant mass due to FSR happens as explained above. Between $M{\rm W}$ W$$M_${\rm W}$$$ and $M{\rm H}$ H$$M_${\rm H}$$-M{\rm W}$ W$$M_${\rm W}$$$ none of the W bosons is resonant, and the contribution to the lowest-order width is small. Therefore, owing to the redistribution of events via photon emission the relative corrections are large in this regime. Below $M{\rm H}$ H$$M_${\rm H}$$-M{\rm W}$ W$$M_${\rm W}$$$, where the other W boson can become resonant, qualitatively the same FSR effects are visible as in the vicinity of the resonance at $M{\rm W}$ W$$M_${\rm W}$$$: apart from a constant positive off-set in the relative corrections, events are distributed from the right to the left of the maximum. Figure \ref{['fig:zinv']} shows the corresponding invariant-mass distributions for the decay ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ with $M{\rm H}$ H$$M_${\rm H}$$=200\,{\rm GeV}$ and $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$. Distribution in the invariant mass of the ${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ pair (l.h.s.) and relative corrections to the distributions in the invariant masses of the ${\rm e^-}$ e^-${\rm e^+}$ e^+$$ and ${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ pairs (r.h.s.) in the decay ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ for $M{\rm H}$ H$$M_${\rm H}$$=200\,{\rm GeV}$ and $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$.The generic features of the plots are similar to the decay into W bosons. For $M{\rm H}$ H$$M_${\rm H}$$=200\,{\rm GeV}$, i.e. above the ZZ threshold, there is a resonance region around $M{\rm Z}$ Z$$M_${\rm Z}$$$, and the corrections become large in the non-collinear-safe case. Photon recombination rearranges the events, so that the fermion logarithms cancel. For Higgs masses below the ZZ threshold, such as for $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$, one Z boson or the other is resonant for $M_{f\bar{f}}\sim M{\rm Z}$ Z$$M_${\rm Z}$$$ or $M_{f\bar{f}}\mathrel{\raisebox{-.3em}{$\stackrel{ <}{\sim}$}} M{\rm H}$ H$$M_${\rm H}$$-M{\rm Z}$ Z$$M_${\rm Z}$$$, respectively. The shape and the large size of the corrections are due to collinear FSR as explained above. In Ref. Choi:2002jk it was pointed out that the kinematical threshold near $M{\rm H}$ H$$M_${\rm H}$$-M{\rm Z}$ Z$$M_${\rm Z}$$$ where the other Z boson can become on shell, which is at $M_{f\bar{f}}\sim80\,{\rm GeV}$ in Figure \ref{['fig:zinv']}, can be used to verify the spin of the Higgs boson. While the rise of the width near this threshold is proportional to $\beta\propto\sqrt{[1-(M{\rm Z}$ Z$$M_${\rm Z}$$+M_{f\bar{f}})^2/M{\rm H}$ H$$M_${\rm H}$$^2][1-(M{\rm Z}$ Z$$M_${\rm Z}$$-M_{f\bar{f}})^2/M{\rm H}$ H$$M_${\rm H}$$^2]}$ for a spin-0 particle, it would be proportional to $\beta^3$ for a spin-1 particle. Figure \ref{['fig:zinv']} shows that the radiative corrections influence the slope at the kinematical threshold. Finally, in Figure \ref{['fig:fsr']} we investigate the influence of the contribution $\delta_{{\rm FSR}}$ of higher-order FSR to the complete relative correction $\delta$ on the invariant-mass distribution of ${\rm \mu^-}$ μ^-$\bar{\nu$ν$_{\mu}}$ and ${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ in the decays ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ and ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$. The invariant mass is defined via the momenta of the fermions alone, i.e. without photon recombination. If photon recombination was applied, the leading logarithmic FSR corrections, as described in Section \ref{['se:fsr']}, would vanish completely. Subtracting the $\cal{O}\alpha)$O(α)$$ terms (\ref{['eq:Oafsr']}) with (\ref{['eq:Oasf']}) from (\ref{['eq:FSR']}) with the structure functions (\ref{['eq:GammaFSR']}) yields the contribution that is beyond $\cal{O}\alpha)$O(α)$$. In Figure \ref{['fig:fsr']} the impact of this contribution is studied revealing corrections of up to $4\%$ in regions where the lowest-order result is relatively small. Figure \ref{['fig:fsr']} also shows the comparison between the structure function with and without the exponentiation of the soft-photon parts in (\ref{['eq:GammaFSR']}) and (\ref{['eq:FSRreexpand']}), respectively. The difference is beyond $\cal{O}\alpha^3)$O(α^3)$$ and turns out to be tiny. Influence of the leading logarithmic terms of FSR on the invariant-mass distribution of ${\rm \mu^-}$ μ^-$\bar{\nu$ν$_{\mu}}$ and ${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ in the decays ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ and ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$. The different curves correspond to the result with exponentiation, without exponentiation, and to the sum of $\alpha^2$ and $\alpha^3$ terms, which are labelled "beyond $\cal{O}\alpha)$O(α)$$". The investigation of angular correlations between the fermionic decay products is an essential means of testing the properties of the Higgs boson. In Refs. Nelson:1986kiChoi:2002jk it was demonstrated how the spin of the Higgs boson can be determined by looking at the angle between the decay planes of the Z bosons in the decay ${\rm H}$ H$\to{\rm Z}$ Z${\rm Z}$ Z$$. This angle can be defined by \cos{\phi'}=\frac{({\bf k}_{12}\times{\bf k}_1)({\bf k}_{12}\times{\bf k}_3)}{|{\bf k}_{12}\times{\bf k}_1||{\bf k}_{12}\times{\bf k}_3|},\mathop{{\rm sgn}}\nolimits(\sin{\phi'})=\mathop{{\rm sgn}}\nolimits\{{\bf k}_{12}\cdot[({\bf k}_{12}\times{\bf k}_1)\times ({\bf k}_{12}\times{\bf k}_3)]\},where ${\bf k}_{12}={\bf k}_1+{\bf k}_2$. The l.h.s. of Figure \ref{['fig:phi']} shows the differential decay width for ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ as a function of $\phi'$ revealing a $\cos{2\phi'}$ term. As was noticed in Refs. Nelson:1986kiChoi:2002jk, this term would be proportional to ($-\cos{2\phi'}$) if the Higgs boson was a pseudo-scalar. Distribution in the angle between the ${\rm Z}$ Z$\to l^-l^+$ decay planes in the decay ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ (l.h.s.) and corresponding relative corrections (with photon recombination) (r.h.s.) for $M{\rm H}$ H$$M_${\rm H}$$=200\,{\rm GeV}$.Note that for non-photonic events the definition of $\phi'$ coincides with the definition given in Ref. Denner:2000bj where ($-{\bf k}_{34}\times{\bf k}_3$) with ${\bf k}_{34}={\bf k}_3+{\bf k}_4$ was used instead of (${\bf k}_{12}\times{\bf k}_3$). Explicitly, $\phi$ was defined by \cos{\phi}=\frac{({\bf k}_{12}\times{\bf k}_1)(-{\bf k}_{34}\times{\bf k}_3)}{|{\bf k}_{12}\times{\bf k}_1||-{\bf k}_{34}\times{\bf k}_3|},\mathop{{\rm sgn}}\nolimits(\sin{\phi})=\mathop{{\rm sgn}}\nolimits\{{\bf k}_{12}\cdot[({\bf k}_{12}\times{\bf k}_1)\times (-{\bf k}_{34}\times{\bf k}_3)]\}.However, this definition yields large negative contributions at $\phi=0^{\circ}$ and $\phi=180^{\circ}$. As was explained in Ref. Denner:2000bj, this is an effect of the suppressed phase space in the real corrections. At $\phi=0^{\circ}$ and $\phi=180^{\circ}$ the phase space for photonic events shrinks to the configurations where the photon is either soft or lies in the decay plane of the gauge bosons. Thus, the negative contributions from the virtual corrections are not fully compensated by the real corrections. Using ${\bf k}_{12}\times{\bf k}_3$ as in (\ref{['eq:phipr']}) avoids this suppression and gives rise to a smooth dependence of the corrections on $\phi$ as can be seen on the r.h.s. of Figure \ref{['fig:phi']} which shows the relative corrections for $\phi$ and $\phi'$ in the decay ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$. Since the difference of $\phi$ and $\phi'$ is only due to photons, this, again, emphasizes the influence of the photon treatment. In contrast to the invariant-mass distribution of Figure \ref{['fig:winv']}, photon recombination does not produce any significant effect for the observables $\phi,\phi'$. This is because adding a soft or collinear photon to a fermion momentum does not change its direction significantly and, thus, has only a small influence on the angles $\phi,\phi'$. The distribution in the decay angle of the ${\rm \mu^-}$ μ^-$$ relative to the corresponding Z boson in the decay ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ is shown in Figure \ref{['fig:th']}. Distribution in the angle between the ${\rm \mu^-}$ μ^-$$ and the corresponding Z boson in the rest frame of the Z boson (l.h.s.) and corresponding relative corrections with and without photon recombination (r.h.s.) in the decay ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ for $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$ and $M{\rm H}$ H$$M_${\rm H}$$=200\,{\rm GeV}$.The angle is defined in the rest frame of the Z boson. Since the Z bosons resulting from Higgs decay are preferably longitudinally polarized, the distribution involves a component proportional to $\sin^2\theta_{{\rm Z}$ Z${\rm \mu^-}$ μ^-$}$. The relative corrections which are shown in the plot on the r.h.s. reveal a strong enhancement in the forward and backward direction if no recombination is applied. This enhancement is due to events where the ${\rm \mu^+}$ μ^+$$ emits a collinear photon and has only a small energy left. Since the momentum of the Z boson is defined via its decay fermions, it has almost the same momentum as the ${\rm \mu^-}$ μ^-$$. This configuration is enhanced by collinear logarithms which are not compensated by virtual contributions. After applying photon recombination, the momentum of the Z boson is defined via the sum of the fermion and photon momenta. Thus, the ${\rm \mu^-}$ μ^-$$ is not necessarily collinear to the Z boson anymore, and events are rearranged to smaller $|\cos\theta_{{\rm Z}$ Z${\rm \mu^-}$ μ^-$}|$ giving rise to a flatter distribution. Next, we consider the distribution in the angle between two fermions. In the case of ${\rm H}$ H$\to{\rm W}$ W${\rm W}$ W$$ the angle between the charged fermions can be used to discriminate the Higgs signal events from background events Dittmar:1996ss, because the fermions are emitted preferably in the same direction. This can be understood as follows. At leading order, the only non-vanishing helicity amplitudes for ${\rm H}$ H$\to{\rm W}$ W${\rm W}$ W$$ are those with equal-helicity W bosons. Since W bosons only couple to left-handed particles and due to angular momentum conservation, particles (anti-particles) are emitted preferably in the forward direction of transverse W bosons with negative (positive) helicity, and anti-particles (particles) in the backward direction. As, close to threshold, 2/3 of the W bosons are transverse and as the W bosons fly in opposite directions, a particle and an anti-particle of their decay products will be emitted preferably in the same direction, resulting in small angles between these particles. In the decay ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ neither the Higgs-boson nor the W-boson momenta can be reconstructed from the decay products. The distribution in the angle between the ${\rm e^+}$ e^+$$ and ${\rm \mu^-}$ μ^-$$ can, thus, only be studied upon including the Higgs-production process. If the Higgs boson was, however, produced without transverse momentum, or if the transverse momentum was known, the angle between ${\rm e^+}$ e^+$$ and ${\rm \mu^-}$ μ^-$$ in the plane perpendicular to the beam axis could be studied without knowledge of the production process. We define the transverse angle between ${\rm e^+}$ e^+$$ and ${\rm \mu^-}$ μ^-$$ in the frame where ${\bf k}_{{\rm H}$ H$,{\rm T}}=0$ as \cos\phi_{{\rm e}{\rm e}\mu,{\rm T}}=\frac{{\bf k}_{2,{\rm T}}\cdot{\bf k}_{3,{\rm T}}}{|{\bf k}_{2,{\rm T}}|{|\bf k}_{3,{\rm T}}|},\mathop{{\rm sgn}}\nolimits(\sin{\phi_{{\rm e}{\rm e}\mu,{\rm T}} })=\mathop{{\rm sgn}}\nolimits\{{\bf e}_z\cdot({\bf k}_{2,{{\rm T}}}\times{\bf k}_{3,{{\rm T}}})\},where ${\bf k}_{i,{\rm T}}$ are the transverse components of the fermion momenta w.r.t. the unit vector ${\bf e}_z$, which could be identified with the beam direction of a Higgs production process. The corresponding distribution, together with the influence of the corrections, is shown in Figure \ref{['fig:phitr']}. Distribution in the transverse angle between ${\rm e^+}$ e^+$$ and ${\rm \mu^-}$ μ^-$$ including corrections (l.h.s.) and corresponding relative corrections (r.h.s.) with and without applying photon recombination in the decay ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$ for $M{\rm H}$ H$$M_${\rm H}$$=140\,{\rm GeV}$ and $M{\rm H}$ H$$M_${\rm H}$$=170\,{\rm GeV}$. The enhancement for small angles, which was explained above, is transferred to the distribution of the transverse angle $\phi_{{\rm e}$ e$\mu,{\rm T}}$. Since the photon recombination does not change the direction of the fermions, it does not have any visible effect on the relative corrections. Finally, we investigate the distribution of the angle between ${\rm e^-}$ e^-$$ and ${\rm \mu^-}$ μ^-$$ in the decay ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$. We prefer to choose the angle between two fermions with the same charge because this constitutes an unambiguous choice in the decay ${\rm H}$ H$\to{\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$. Figure \ref{['fig:th13']} shows the tendency that the fermions are emitted in opposite directions for the same reason as explained above. Distribution in the angle between ${\rm e^-}$ e^-$$ and ${\rm \mu^-}$ μ^-$$ including corrections (l.h.s.) and corresponding relative corrections (r.h.s.) with and without applying photon recombination in the decay ${\rm H}$ H$\to{\rm e^-}$ e^-${\rm e^+}$ e^+${\rm \mu^-}$ μ^-${\rm \mu^+}$ μ^+$$ for $M{\rm H}$ H$$M_${\rm H}$$=200\,{\rm GeV}$.However, this feature is not as pronounced as in ${\rm H}$ H$\to\nu$ν$_{{\rm e}}{\rm e^+}$ e^+${\rm \mu^-}$ μ^-$\bar{$$\nu_{\mu}$, because Z bosons do also couple to right-handed fermions so that one Z boson might decay into a left-handed fermion and the other into a right-handed fermion. The radiative corrections tend to reduce the enhancement in forward direction and do not depend on photon recombination. The decays of the Standard Model Higgs boson into four leptons via a W-boson or Z-boson pair lead to experimental signatures at the LHC that are both important for the search for the Higgs boson and for studying its properties. To exploit this possibility a Monte Carlo event generator for the decays ${\rm H}$ H$\to{\rm W}$ W${\rm W}$ W$/{\rm Z}$ Z${\rm Z}$ Z$\to4\,$leptons is needed that properly accounts for the relevant radiative corrections, in order to achieve the necessary precision in predictions. Prophecy4f is an event generator dedicated to this task. We have shown first results of this generator and described the underlying calculation. In detail, we have presented the complete electroweak radiative corrections of ${\cal O}(\alpha)$ to the decays ${\rm H}$ H$\to4\,$leptons, supplemented by corrections beyond ${\cal O}(\alpha)$ originating from heavy-Higgs effects and final-state radiation. The intermediate W- and Z-boson resonances are treated in the so-called complex-mass scheme, which fully preserves gauge invariance and does not employ any type of expansion or on-shell approximation for the intermediate gauge-boson resonances. Consequently, the calculation is equally valid above, in the vicinity of, and below the WW and ZZ thresholds. The corrections to partial decay widths typically amount to some per cent and increase with growing Higgs mass $M{\rm H}$ H$$M_${\rm H}$$$, reaching about 8% at $M{\rm H}$ H$$M_${\rm H}$$\sim500\,{\rm GeV}$. This statement, however, applies only if the lowest-order decay widths are already evaluated with the full off-shell effects of the intermediate W and Z bosons, in particular near and below the WW and ZZ thresholds. The on-shell (narrow-width) approximation for the corrections is good within $0.5{-}1\%$ of the width for Higgs masses sufficiently above the corresponding gauge-boson pair threshold, as long as the lowest-order prediction consistently includes the off-shell effects of the gauge bosons. For ${\rm H}$ H$\to{\rm W}$ W${\rm W}$ W$\to4f$ the narrow-width approximation fails by about 10% for Higgs masses that are only $2\,{\rm GeV}$ above the ${\rm W}$ W${\rm W}$ W$$ threshold, because the instability of the W bosons significantly influences the Coulomb singularity near threshold. Only a calculation that keeps the full off-shellness of the W and Z bosons can describe the threshold regions properly. We have given a simple improved Born approximation for the partial widths that reproduces the full calculation within $\mathrel{\raisebox{-.3em}{$\stackrel{ <}{\sim}$}}2\%$ for Higgs masses below $400\,{\rm GeV}$. In this regime our complete calculation has a theoretical uncertainty below 1%. For larger Higgs masses we expect that unknown two-loop corrections that are enhanced by $G\mu$G_μ$M{\rm H}$ H$$M_${\rm H}$$^2$ deteriorate the accuracy. Finally, for $M{\rm H}$ H$$M_${\rm H}$$\mathrel{\raisebox{-.3em}{$\stackrel{ >}{\sim}$}}700\,{\rm GeV}$ it is well known that perturbative predictions become questionable in general. For angular distributions, which are important in the verification of the discrete quantum numbers of the Higgs boson, the corrections are of the order of $5{-}10\%$ and distort the shapes. For invariant-mass distributions of fermion pairs, which are relevant for the reconstruction of the gauge bosons, the situation is similar to gauge-boson pair production processes such as ${\rm e^+}$ e^+${\rm e^-}$ e^-$\to{\rm W}$ W${\rm W}$ W$\to4\,$fermions, i.e. the corrections can reach several tens of per cent depending on the treatment of photon radiation. In its present version the Monte Carlo event generator Prophecy4f deals with fully leptonic final states, a situation most relevant for the LHC. The generalization to semi-leptonic and hadronic final states, including a proper description of QCD corrections, will be described in a forthcoming publication.We thank M. Spira for helpful discussions about Hdecay.