Higgs pseudo observables and radiative corrections
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Abstract
We show how leading radiative corrections can be implemented in the general description of \(h\rightarrow 4\ell \) decays by means of pseudo observables (PO). With the inclusion of such corrections, the PO description of \(h\rightarrow 4\ell \) decays can be matched to nexttoleadingorder electroweak calculations both within and beyond the Standard Model (SM). In particular, we demonstrate that with the inclusion of such corrections the complete nexttoleadingorder SM prediction for the \(h\rightarrow 2e2\mu \) dilepton mass spectrum is recovered within \(1\,\%\) accuracy. The impact of radiative corrections for nonstandard PO is also briefly discussed.
Keywords
Invariant Mass Radiative Correction Effective Field Theory Standard Model Prediction Dilepton Invariant Mass1 Introduction
The decays of the Higgs particle, h(125), can be characterised by a set of pseudo observables (PO) that describes, in great generality, possible deviations from the Standard Model (SM) in the limit of heavy New Physics (NP) [1].
Soft and collinear photon emission represents a universal correction factor [2, 3] that can be implemented, by means of appropriate convolution functions (or, equivalently, showering algorithms such as those adopted in PHOTOS [4], PYTHIA [5], or SHERPA [6]) irrespective of the specific shortdistance structure of the amplitude.^{1} In this paper we illustrate how this works, in practice, in the \(h\rightarrow 4\ell \) case.
We focus our analysis to the \(h\rightarrow 2e 2\mu \) case, that is particularly interesting for illustrative purposes: the effect of radiative corrections can be implemented by simple analytic formulae, allowing a transparent comparison with numerical methods. As we will show, the inclusion of the universal QED corrections is necessary and sufficient to reach an accurate theoretical description of the Higgs decay spectrum, that recovers the best uptodate SM predictions in absence of NP.
2 QED corrections for the \(h\rightarrow 4\ell \) dilepton spectrum
Working in the limit of massless leptons, we need to introduce two independent IR regulators for soft and collinear divergences. We choose them to be: (i) the minimal fraction of invariant mass lost by the dilepton invariantmass system; (ii) the minimal invariant mass of a single lepton plus (collinear) photon (\(m^*\)).
We have determined the structure of \(\omega _1\) by means of an explicit \(O(\alpha )\) calculation of the real emission, while \(\omega _2\) has been determined by the condition \(\int _0^1 \text {d}x \omega (x,x_*) =1\). The latter condition implies a redefinition of \(O(\alpha /\pi )\), not enhanced by large logs, of the PO characterizing the nonradiative amplitude.
3 Comparison with full NLO electroweak corrections
In this section we present a comparison of the SM predictions for the \(h\rightarrow 2e 2\mu \) dilepton invariant mass spectrum obtained using full NLO electroweak corrections [10], and the PO decomposition “dressed” with leading QED corrections, as described above.
The complete SM NLO electroweak corrections to \(h\rightarrow 4\ell \) have been computed in [10], and the results have been implemented in the Monte Carlo event generator Prophecy4f [11]. We have used Prophecy4f version 2.0 to generate 200 millions weighted events for the recombination mass parameter \(m_* =1\) GeV. We have used the default Prophecy4f SM inputs except for setting the Higgs boson mass to 125 GeV. Prophecy4f adopts the dipole subtraction formalism [12] for the treatment of soft and collinear divergences, and the socalled “photonrecombination” is applied. In particular, if the invariant mass of a lepton and a photon is smaller than \(m_*\), the photon momentum is added to the lepton momentum [10]. As a result, \(m_*\) coincides with the collinear cutoff introduced in the previous section.

The spectrum obtained with the PO decomposition of the amplitude, “dressed” with leading QED corrections, provides an excellent approximation (within 1 % accuracy) to the spectrum obtained with full NLO EW corrections.^{2}

The effect of the leading QED corrections can be large, exceeding \(10\,\%\) in specific regions of the phase space. It therefore must be included, in view of a precise datatheory comparison, also when fitting beyondtheSM parameters.

The PO “dressed” spectrum is obtained setting \(\epsilon _i=0\) (i.e. to their LO SM values). The good agreement with the complete NLO calculation confirms that the \(O(\alpha /\pi )\) redefinition of the \(\epsilon _i\) is a small effect, with no observable consequences for the \(h\rightarrow 2e2\mu \) dilepton invariant mass spectrum.
4 Implications for New Physics
As shown in Fig. 1 (left), radiative corrections can be sizable and must be included also when going beyond the SM. Having demonstrated the validity of our QED improved predictions to describe such effects, we are in position to apply the method in the presence of an arbitrary New Physics contribution to \(h\rightarrow 2e 2\mu \) decay as parameterised by generic PO [1]. As an illustrative example, we consider the impact of the leading QED corrections for nonstandard values of \(\kappa _{ZZ}\), \(\epsilon _{Ze_L}\), \(\epsilon _{Ze_R}\), \(\epsilon _{Z\mu _L}\), and \(\epsilon _{Z\mu _R}\).
To draw some general conclusions we analyse three benchmark points, chosen such that the deviations of the total \(h\rightarrow 2e 2\mu \) decay rate from the SM prediction are always small,^{3} but the impact on the spectrum are quite different. The results of the inclusion of QED corrections are shown in Fig. 1 (right). As in the left panels, we plot the dilepton invariant mass distribution normalized to the total rate (upper plot) and the ratio between NLO and LO (lower plot).

Benchmark I [\(\kappa _{ZZ}=1.3\), \(\epsilon _{Ze_L}=\epsilon _{Z\mu _L}=0.05\), \(\epsilon _{Ze_R}=\epsilon _{Z\mu _R}=0.05\) (dot–dashed blue)]. Here the deviation from the SM point in the Higgs PO parameter space is small: this benchmark point is compatible with naive power counting in the linear EFT expansion. As a consequence, small deformations in the spectrum are obtained (upper panel) and the relative QED corrections are SMlike (lower panel). In this regime, the leading QED corrections can be directly extracted from the SM result (via an appropriate NLO/LO reweighting).

Benchmark II [\(\kappa _{ZZ}=0\), \(\epsilon _{Ze_L}=\epsilon _{Z\mu _L}=0.26\), \(\epsilon _{Ze_R}=\epsilon _{Z\mu _R}=0\) (dotted red)]. Here the deviation from the SM point is sizable, beyond the naive power counting within a generic EFT (both linear and nonlinear). However, the PO configuration is such that the deviations from the SM in the spectrum are small. This implies that the relative impact of QED corrections is still SMlike.

Benchmark III [\(\kappa _{ZZ}=0.3\), \(\epsilon _{Ze_L}=0.45\) and \(\epsilon _{Z\mu _L}=\epsilon _{Ze_R}=\epsilon _{Z\mu _R}=0\) (solid green)]. In this example we observe a sizable distortion of the dilepton shape (upper panel). As a consequence, the relative impact of the QED corrections is quite different from the SM case (a description of radiative corrections by NLO/LO reweighting of the SM result would not provide a good approximation).
5 Conclusions
The dominant electroweak corrections to \(h\rightarrow 4\ell \) decays are due to the universal soft and collinear photon emission. As shown in Fig. 1, these can lead to distortions of the dilepton invariant spectrum of \(O(10\,\%)\) is specific regions of the phase space. These effects are of the same order as the expected modifications from the SM under the assumption of underlining linear EFT [13]. It is then mandatory to properly incorporate these corrections in a consistent way both within and beyond the SM.
As we have shown in this paper, this can be achieved in general terms within the framework of the Higgs PO. In particular we have shown that: (i) the QED corrected predictions for the \(h\rightarrow 2e2\mu \) dilepton invariant mass spectra, with PO fixed to their SM LO values, are in agreement with the full NLO electroweak SM predictions within \(1\,\%\) accuracy; (ii) the QED corrections in the presence of NP can be sizable and significantly different from the SM case.
Footnotes
 1.
For a discussion about the implementation of universal QED corrections in a general EFT context see also Ref. [7].
 2.
The \(\sim \)2 % deviations at the border of the phase space are expected due the breakdown of the approximation \(m_{\ell \ell } \gg m_*\) employed in the analytic evaluation of the radiation function.
 3.
The dependence of the total rate on the PO can be found in Ref. [13].
Notes
Acknowledgments
This research was supported in part by the Swiss National Science Foundation (SNF) under Contract 200021159720.
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