# Electroweak corrections to the fermionic decays of heavy Higgs states

## Abstract

Extensions of the standard model often come with additional, possibly electroweakly charged Higgs states, the prototypal example being the Two-Higgs-Doublet Model. While collider phenomenology does not exclude the possibility for some of these new scalar fields to be light, it is relatively natural to consider masses in the multi-TeV range, in which case the only remaining light Higgs boson automatically receives SM-like properties. The appearance of a hierarchy between the new-physics states and the electroweak scale then leads to sizable electroweak corrections, e. g. in the decays of the heavy Higgs bosons, which are dominated by effects of infrared type, namely Sudakov logarithms. Such radiative contributions obviously affect the two-body decays, but should also be paired with the radiation of electroweak gauge bosons (or lighter Higgs bosons) for a consistent picture at the one-loop order. Resummation of the leading terms is also relatively easy to achieve. We re-visit these questions in the specific case of the fermionic decays of heavy Higgs particles in the Next-to-Minimal Supersymmetric Standard Model, in particular pointing out the consequences of the three-body final states for the branching ratios of the heavy scalars.

## 1 Introduction

Many extensions of the standard model (SM) imply the existence of an extended Higgs sector. While the reality of a SM-like Higgs boson is firmly established by the experiments at the Large Hadron Collider (LHC) [1, 2, 3], the status of other Higgs states remains largely speculative. At the moment, the absence of conclusive signals for such new states has only limited implications since, in many models, only a marginal portion of the parameter space has been actually tested. In fact, light electroweakly-charged scalar particles with a mass below that of the top quark continue to be phenomenologically viable [4], even though such scenarios receive constraints from multiple directions. The situation is much looser for singlet-dominated states, such as predicted in the Next-to-Minimal Supersymmetric Standard model (NMSSM) for instance (see e. g. Ref. [5] for a discussion of the constraints from Run I at the LHC). On the other hand, it is tempting to retreat to the (multi-)TeV scale for the mass of the new Higgs bosons, because then the SM-like characteristics of the observed state (see Refs. [6, 7, 8]) are almost automatically fulfilled, due to the decoupling properties of the heavy Higgs particles (under the assumption of perturbative couplings). In such a case, however, the presence of a comparatively high scale introduces a hierarchy with respect to the electroweak interactions that could lead to large radiative corrections, typically in the form of Sudakov logarithms – see e. g. Refs. [9, 10, 11, 12] for a few related references.

Below, we specialize in the decays of heavy Higgs bosons in the particular case of the NMSSM, although our analysis can be easily extended to any other model including additional Higgs states, in particular models based on a Two-Higgs-Doublet Model (THDM) of type II. The NMSSM [13, 14] – as any supersymmetric (SUSY) [15, 16] extension of the SM – shields the mass of the SM-like Higgs against large radiative corrections from new physics at high-energy scales (e. g. the GUT or Planck scale), suggesting a technically natural answer to the ‘Hierarchy Problem’. Other motivations are the resolution of the ‘\(\mu \) problem’ [17] or the rich phenomenology of the Higgs or neutralino sectors, see e. g. [5, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] for a few recent discussions. The NMSSM contains one pair of charged and four additional neutral Higgs bosons beyond the SM-like state, involving two new \({\mathcal{CP}}\)-even and two \({\mathcal{CP}}\)-odd degrees of freedom. Several public tools propose an evaluation of the two-body Higgs decays. The standard has long been a QCD-improved calculation: the decay routines of NMSSMTools [36, 37, 38, 39] and NMSSMCALC [40, 41] are based on HDECAY [42, 43]; SOFTSUSY [44, 45, 46] also performs at the same order. More recently, full one-loop analyses of the two-body Higgs decays have been performed with the code SloopS [47, 48, 49, 50] or in the \(\overline{\mathrm {DR}}\) \(\left( \overline{\mathrm {MS}}\right) \) scheme for generic models [51] with the code SPHENO [52, 53, 54, 55], which employs SARAH [56, 57, 58, 59]. Also in Ref. [60] the two-body Higgs decays into SM final states were considered at the full one-loop order. Similar projects have been presented for the THDM and its extensions [61, 62, 63]. Yet – with the exception of photon and gluon radiation as well as the production of off-shell gauge bosons below threshold that subsequently decay into fermions (see e. g. Ref. [64] for a recent summary in the MSSM) – little attention has been paid to the three-body decays, which, however, intervene at the same order as the one-loop corrections to two-body decays.

The degeneracy among the states forming an \(SU(2)_{\text {L}}\) multiplet is lifted by the electroweak symmetry breaking, so that the mass-squared differences among the partners of a scalar doublet are expected to be of the order of \(M_Z^2\). Therefore, in scenarios of a THDM where one doublet mass is much larger than the electroweak scale, actually four Higgs bosons have almost degenerate large masses and approximately organize as an \(SU(2)_{\text {L}}\) doublet, while one Higgs boson is automatically endowed with SM-like properties, thus making it a good candidate for explaining the signals observed by the ATLAS and CMS collaborations – provided its mass falls within the suitable interval at \({\mathord {\sim }}\,125\) GeV. This setup fulfills the conditions of the decoupling limit [65] and the decays of the heavy Higgs bosons into SM final states tend to be dominated by the fermionic channels: the couplings of the heavy states to a pair of electroweak gauge bosons vanish (approximately for the \({\mathcal{CP}}\)-even Higgs), because the doublet formed by the heavy states is orthogonal to that generating the electroweak-symmetry-breaking vacuum expectation value (v.e.v.); in addition, the decays into a SM-like Higgs plus an electroweak gauge boson are also suppressed, because the decoupling states are approximately partners of one another under the electroweak gauge group, and not of the SM-like Higgs;^{1} finally, Higgs-to-Higgs decays are suppressed in the limit of heavy initial states – the triple Higgs coupling of electroweak size implies a suppression \({\mathord {\sim }}\,M_Z/M_h\), where \(M_h\) represents the mass of the initial Higgs boson. On the other hand, at least some of the fermionic decays are expected to be unsuppressed, depending on the type of THDM. For a type II framework, the standard search channels at the LHC thus involve tau-onic final states, while the \(b\bar{b}\) or \(t\bar{t}\) decays are actually sizable but suffer from the QCD background. The decays of singlet-dominated states are more difficult to characterize generically. A pure singlet could only decay radiatively into SM final states (and even not at all for \(\lambda =0\)). In general, the decays of mostly singlet-like states into SM particles are thus dominated by their subleading doublet components – therefore their bosonic decays, e. g. into light Higgs bosons, are not necessarily suppressed. Below, we focus on the fermionic decays for simplicity, but the procedure can be generalized to bosonic final states as well.

The main purpose of this paper consists in analyzing the electroweak corrections to the two-body fermionic decays of heavy Higgs bosons, and their interplay with three-body decays involving the radiation of an electroweak gauge boson. The noteworthy difference compared to the case of QED and QCD corrections is that the radiation of massive electroweak gauge bosons leads to clearly distinguishable final states, as opposed to soft and collinear photons and gluons. The corresponding decays could thus be measured separately and there is no justification on the theoretical side for considering only inclusive decay widths (summing over two-body decays and the corresponding ones with radiated *W*, *Z*): therefore the potentially large Sudakov double logarithms are experimentally accessible effects that are expected to reduce the branching fractions of the two-body decays. In any case, from order-counting it appears mandatory to take the three-body final states into account if one wants to perform a consistent analysis of the branching ratios for two-body decays of heavy Higgs bosons at the full one-loop order. To our knowledge, such effects have not been considered in the NMSSM (or the MSSM) yet, and we expose here how we implement these channels in view of a future inclusion within a version of FeynHiggs [70, 71, 72, 73, 74, 75, 76, 77] dedicated to the NMSSM [60, 78, 79].

In the following section, we discuss the formal aspects of our evaluation of the two- and three-body decays, emphasizing the possibility to capture most of the electroweak corrections within a simple resummation of Sudakov double logarithms [80]. We then illustrate the numerical impact of the electroweak corrections in a scenario with heavy SUSY sector and focusing on doublet-dominated Higgs bosons in the initial state. Finally, Sect. 4 summarizes our achievements.

## 2 Higgs decays into SM fermions in the NMSSM

In this section, we summarize the main theoretical ingredients entering the decays of heavy Higgs bosons into SM fermions. We follow the notations of Refs. [60, 79].

### 2.1 Two-body decay width

#### 2.1.1 Decay of the Higgs bosons at one-loop order: automated calculation

We rely on the automated calculation of FeynArts [81, 82], FormCalc [83] and LoopTools [83], using the model file (and renormalization scheme) presented in Ref. [79].

External Higgs fields are upgraded to on-shell fields using the transformation matrix \(Z^{\mathrm{mix}}\) that is determined by the LSZ reduction, see Ref. [79]. The mixing of the external Higgs leg with the electroweak neutral current is processed diagrammatically, and we subtract gauge-violating effects of two-loop order that appear due to the difference between the (kinematical) loop-corrected Higgs mass and the tree-level mass, see Ref. [60]. The fermion fields are renormalized on-shell.

- The virtual QCD and QED corrections are processed separately and combined with the real-radiation contributions in order to define an infrared (IR)-finite correction factor [67, 68, 84] to the pure born width \(\Gamma ^{\text {Born}}\) describing the inclusive fermionic width (with radiated gluons and photons in the final state). The QCD logarithms are absorbed within the definition of running Yukawa couplings evaluated at the scale of the decaying Higgs:where \(Y_f\) represents the relevant Yukawa coupling absorbing QCD logarithms; \(c^{\text {QCD}}\) and \(c^{\text {QED}}\) represent the QCD and QED correction factors – see e. g. Eq. (2.12) of Ref. [60].$$\begin{aligned} \frac{\Gamma ^{\text {QCD+QED}}}{\Gamma ^{\text {Born}}}&= \left( \frac{Y_f{(M_h)}}{Y_f{(m_t)}}\right) ^2 \left[ 1+c^{\text {QCD}}+c^{\text {QED}}\right] \end{aligned}$$(2.1)
The leading SUSY corrections amount to contributions to effective dimension 4 Higgs–fermion operators. They are explicitly extracted in the

*SU*(2)-conserving limit and re-arranged so as to provide a resummation of the \(\tan \beta \)-enhanced effects, see Refs. [40, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97]; a linearized (non-resummed) version is subtracted from the actual one-loop diagrammatic calculation in order to avoid double counting.The full one-loop corrections (excluding the QED and QCD corrections) to the Higgs–fermion vertex are derived automatically from our model file. However, we subtract gauge-violating effects of two-loop order through a re-definition of the Higgs–Goldstone couplings by a shift of one-loop order, as explained in section 2.1 of Ref. [60].

The external charged Higgs field and its mass are renormalized on-shell, thus requiring no normalization factor. The mixing with the electroweak charged current is processed diagrammatically.

- The QCD and QED corrections are factorized out and combined with the real radiation process. The QCD-correction factor is well-known [98]. The QED contribution is somewhat more involved, due to the initial state being electrically charged. We compute this correction factor by considering separately the virtual piece as well as the soft and hard radiation. For the hard radiation, we explicitly take the limit \(m_b\rightarrow 0\). For simplicity, we only provide the leading logarithmic terms for \(H^-\rightarrow b\bar{t}\) below:$$\begin{aligned}&\frac{\Delta \Gamma ^{\text {QED}}}{\Gamma ^{\text {Born}}}[H^-\rightarrow b\bar{t}\,]\nonumber \\&\quad = \frac{e^2}{16\,\pi ^2}\left[ C^{\text {virt}}+C^{\text {soft}}+C^{\text {hard}}\right] , \end{aligned}$$(2.2a)$$\begin{aligned}&C^{\text {virt}}=2 \Bigg [ Q_b^2\,\ln \frac{m_b^2}{M_{H^{\pm }}^2} + Q_t^2\,\ln \frac{m_t^2}{M_{H^{\pm }}^2} \nonumber \\&\qquad \qquad + 2\left( 1+Q_b\,Q_t\right) \Bigg ] \ln \frac{M_{H^{\pm }}^2}{m_{\gamma }^2}, \end{aligned}$$(2.2b)$$\begin{aligned}&C^{\text {soft}}=2 \Bigg [ Q_b^2\,\ln \frac{m_b^2}{M_{H^{\pm }}^2} + Q_t^2\,\ln \frac{m_t^2}{M_{H^{\pm }}^2} \nonumber \\&\qquad \qquad + 2\left( 1+Q_b\,Q_t\right) \Bigg ] \ln \frac{m_{\gamma }^2}{\Omega ^2_{\text {IR}}}\nonumber \\&\qquad \qquad -Q_b^2 \left[ \ln ^2\frac{m_b^2}{M_{H^{\pm }}^2} + 2\,\ln \frac{m_b^2}{M_{H^{\pm }}^2}\right] \nonumber \\&\qquad \qquad -Q_t^2 \left[ \ln ^2\frac{m_t^2}{M_{H^{\pm }}^2} + 2\,\ln \frac{m_t^2}{M_{H^{\pm }}^2}\right] , \end{aligned}$$(2.2c)where$$\begin{aligned}&C^{\text {hard}}=2 \Bigg [ Q_b^2\,\ln \frac{m_b^2}{M_{H^{\pm }}^2} + Q_t^2\,\ln \frac{m_t^2}{M_{H^{\pm }}^2} \nonumber \\&\qquad \qquad + 2\left( 1+Q_b\,Q_t\right) \Bigg ] \ln \frac{\Omega ^2_{\text {IR}}}{M_{H^{\pm }}^2}\nonumber \\&\qquad \qquad +3\left[ Q_b^2\,\ln \frac{m_b^2}{M_{H^{\pm }}^2} + Q_t^2\,\ln \frac{m_t^2}{M_{H^{\pm }}^2}\right] , \end{aligned}$$(2.2d)
*e*represents the electric charge, \(Q_{t,b}\) the fermion charges, \(m_{\gamma }\) the photon-mass regulator, and \(\Omega _{\text {IR}}\) the IR cut on the photon energy. Non-QCD and non-QED one-loop diagrams are calculated with our automated procedure.

^{2}In this procedure, however, also the SUSY contributions to the Higgs wave function – i. e. intervening in the normalization of the \(Z^{\text {mix}}\) matrix – need to be extracted accordingly. This effective tree-level width, including SUSY corrections, is later denoted as \(\Gamma ^{\text {eff}}\). Then, in Ref. [60], we kept the term of the squared one-loop amplitude in the definition of the width: this was justified in the case where one-loop contributions dominate over the tree-level result. However, the electroweak corrections that we study below are essentially proportional to the tree-level amplitude, meaning that they would be suppressed together with it. In this case, the full one-loop squared term is actually misleading, since its impact can be substantially altered by the inclusion of the interference of the amplitude of two-loop order with the tree-level one. We thus discard this term, keeping only a squared one-loop amplitude for those contributions that are not proportional to the tree-level term (and subleading in the channels that we are considering).

#### 2.1.2 Electroweak double logarithms

- 1.
the well-known fermion–fermion–vector triangle:

the associated loop-function contributes \({\mathord {\sim }}\,\frac{1}{2}\ln ^2\left( M_V^2/M_h^2\right) \) at the double-logarithmic order;

- 2.
the fermion–scalar–vector triangle:

under the condition that the internal scalar is almost degenerate with the external Higgs, the loop function contributes \({\mathord {\sim }}\,{-}\frac{1}{4}\ln ^2{\left( M_V^2/M_h^2\right) }\); here, ‘almost degenerate’ covers a range larger than \(M_h\pm M_V\), but much narrower than \(\left[ M_h/2,\,2\,M_h\right] \).

*I*,

*Y*,

*Q*represent the isospin, hypercharge and electric charge of the external state, \(t_{\text {w}}\) and \(s_{\text {w}}\) denote the tangent and sine of the electroweak mixing angle; the \(Q^2\) term subtracts the QED contribution that is considered separately. The reason why this formula fails is simply that one of the (electroweakly charged) external states, the Higgs line, is massive, actually determining the center-of-mass energy.

*t*,

*b*refer to the pieces of the width that are proportional to the squared Yukawa couplings \(Y_{t,\,b}^2\)):

For a singlet-dominated state, the situation is somewhat more subtle. If the mass of this Higgs boson is far from that of other Higgs states, then the fermion–scalar–vector triangle does not produce double-logarithms for lack of a degenerate electroweak partner of the external state. In this case, the double-logarithmic terms coincide with the formula of Eq. (2.3), applied to the fermionic final states (one being left-handed, the other right-handed). However, if the singlet finds itself accidentally in the window of mass degeneracy with the doublet states, then the fermion–scalar–vector topology is relevant, together with possibly substantial doublet-singlet mixing.

#### 2.1.3 Single logarithms

#### 2.1.4 ‘Improved’ width

### 2.2 Real radiation

#### 2.2.1 Full calculation

From the perspective of order-counting, the three-body decay widths at the tree level are of the same order as one-loop corrections to the two-body decays. For a full one-loop evaluation of the total widths (and branching ratios) of the Higgs bosons, it is thus mandatory to include these channels in the analysis. If we restrict ourselves to non-SUSY final states, the only possible relevant three-body final states involve the radiation of gauge bosons or light Higgs bosons. The radiation of photons and gluons was actually already taken into account above, hence included within the two-body decay widths.

Using FeynArts and FormCalc, we perform the calculation of the widths at the tree-level order for the following channels: \(h_i\rightarrow Zf\bar{f}\), \(H^{\pm }\rightarrow W^{\pm }f\bar{f}\), \(h_i\rightarrow h_jf\bar{f}\) for \(f\in \{b,t,\tau \}\), and \(h_i\rightarrow W^-f\bar{f'}\), \(h_i\rightarrow W^+f'\bar{f}\), \(h_i\rightarrow H^-f\bar{f'}\), \(h_i\rightarrow H^+f'\bar{f}\), \(H^+\rightarrow Zf\bar{f'}\), \(H^+\rightarrow h_jf\bar{f'}\) for \((f,f')\in \{(t,b),(\tau ^+,\nu _{\tau })\}\). We stress that contrarily to the radiation of photons and gluons, these final states are clearly distinguishable from the fermionic two-body channels, so that there is no deep reason to consider (only) inclusive decay widths from a theoretical perspective.

The kinematical integral can be performed (e. g.) in the rest frame of the fermion pair in the final state. There, only two steps of integration are non-trivial, one applying to an ‘angular’ variable, the other to the invariant mass-squared *s* of the fermion pair. We perform the angular integral analytically, while that on *s* is carried out numerically. The internal lines are systematically regularized by inserting the total width \(\Gamma \) of the corresponding particles in the propagators. These ‘internal’ widths are calculated from the sum of all relevant two-body processes at the tree level.

Having set up the calculation in this way, the leading terms in the squared amplitudes scale like \(\Gamma {\left[ h\rightarrow Vf\bar{f}'\right] }/\Gamma {\left[ h\rightarrow f\bar{f}'\right] }\propto M_h^2/M_V^2\). This would lead to the unphysical situation where the three-body decay widths grow faster than the two-body widths, and eventually dominate the decays. However, these contributions linear in \(M_h^2/M_V^2\) sum up to zero as long as the heavy Higgs doublets decouple from the electroweak sector. For the sake of not spoiling the physical interpretation, it is thus paramount to preserve these decoupling properties of the heavy Higgs states at the level of the loop-corrected Higgs-mixing matrix.

#### 2.2.2 On-shell contributions

When an internal line can be exchanged on-shell, the corresponding contribution near the pole (after dismissing the quadratically growing terms discussed above) is dominated by the narrow-width approximation. In this case, a piece of the three-body width is already accounted for at the level of the two-body widths. Therefore, we subtract such on-shell contributions explicitly. This step is performed by evaluating the two-body widths and branching ratios that intervene in the considered process at the tree level. It is crucial that the full widths that are employed as regulators in the internal lines are computed at exactly the same order as these intermediate two-body widths.

*M*: this can be easily understood by comparing the lorentzian distribution and its limit for \(\Gamma \rightarrow 0\),

In addition, we expect the radiation of electroweak gauge bosons and Higgs bosons to produce Sudakov double and single logarithms again, when compared to the corresponding two-body width at the Born level. It is useful to extract them explicitly.

#### 2.2.3 Double logarithms

*W*and

*Z*) or the total one.

*W*and

*Z*bosons, as well as additional photon emission at higher orders related to the mixing of abelian and non-abelian parts in the electroweak gauge group (see Ref. [80]), that is associated with the two-body width \(\Gamma ^{\text {Born}}\).

#### 2.2.4 Single logarithms

the logarithms associated to QED contributions, corresponding to the terms in \(e^2\), are processed in the QED-correction factor and are thus subtracted from the full electroweak contribution;

the logarithms associated to the Higgs wave function exactly cancel the electroweak single logarithms from the vertex corrections to the two-body decays: we recover the logarithms of the RGEs only if we subtract the logarithms from the Higgs field normalization (the latter correspond to the RGEs of the Higgs fields).

The radiation of a light singlet by a heavy doublet state is dominated by the topology with a triple-Higgs coupling, with the internal line possessing a mass close to that of the decaying state. The couplings between heavy doublet states and a singlet are typically of the order of \(\sqrt{2}\,\lambda \,\mu _{\text {eff}}\). Thus the radiation of a singlet-like Higgs provides an unsuppressed logarithm if this coupling fulfills the condition \(\sqrt{2}\,\lambda \,\mu _{\text {eff}}=\mathcal {O}{\left( M_{H^{\pm }}\right) }\).

#### 2.2.5 Inclusive three-body width

The inclusive width for electroweak radiation defined in Eq. (2.13) relies on several assumptions that restrict its validity. In particular, the kinematics does not allow for the radiation of an infinite number of weak gauge bosons, meaning that the exponential series of double-logarithmic terms is in fact explicitly truncated. Yet, in the considered regime the mismatch between the truncated series and the exponential is negligible. As we will show in the following section, the double-logarithmic effects amount to \({\mathord {\sim }}\,10\%\) for states at 1 TeV and \({\mathord {\sim }}\,40\%\) at 100 TeV: comparing \(\exp {(0.1)}\) and \(\exp {(0.4)}\) with their Taylor expansion at 0, we observe that the difference remains at or below the permil level already after including the third order, whereas the kinematics allows for a much larger number of final states with electroweak mass. The description of radiated gauge bosons as actual final states forms another limitation, as *W* and *Z* in fact decay into fermions. The ‘genuine’ process is therefore more complex and includes further topologies, beyond those obtained from radiated *W* and *Z* bosons. We believe, however, that this simplified representation still captures the leading contributions to the Higgs decays beyond two-body final states, since, in the considered regime, we do not expect larger effects than the double logarithms, the latter being intimately associated with gauge-boson radiation.

## 3 Numerical analysis

**Two-body decay width**We first consider the decay width into final states including (anti)quarks of the third generation for heavy doublet-like states in the scenario presented above. In Fig. 3, we plot these decay widths against the mass of the charged Higgs \(M_{H^{\pm }}\) (we recall that \(M_{H^{\pm }}\approx M_{h_2}\approx M_{h_4}\)) in several approximations. The QCD and QED corrections are included per default, as well as the decoupling SUSY effects. For the blue curve, we only consider the width resulting from this level of approximation (denoted as \(\Gamma ^{\text {eff corr}}=\Gamma ^{\text {eff}}\,\Gamma ^{\text {QCD+QED}}/\Gamma ^{\text {Born}}\) in the following): this is essentially the popular description employed in public codes such as HDECAY. The green solid curve corresponds to the ‘improved tree-level’ result, where Sudakov double logarithms are included:

As expected, the impact of the electroweak corrections reaches \(\mathcal {O}{\left( 10\%\right) }\) for \(M_{H^{\pm }}\sim 1\) TeV and more than \(40\%\) at \(M_{H^{\pm }}\sim 100\) TeV. While the tree-level-improved width, including the resummed double logarithms (green solid curve), gives a good qualitative approximation, it is still \({\mathord {\sim }}\,5\%\) off as compared to the full result. The agreement can be further refined if we account for the single logarithms and constant terms (green dashed curve; considering only the single logarithms would lead to an approximation of intermediate quality). Obviously, the resummation of Sudakov double logarithms only yields an appreciable effect beyond \(M_{H^{\pm }}{{\,\mathrm{\gtrsim }\,}}10\) TeV, as the red and orange curves remain comparatively close below this mark. It only amounts to \(\mathcal {O}{\left( 5\text {--}10\%\right) }\) as compared to \(\Gamma ^{\text {1L}}\) beyond.

**Three-body widths** Now, we turn to the three-body decays. In the upper row of Fig. 4, we show the widths of the three-body channels \(h_2\rightarrow W^-t\bar{b}\) (left) and \(H^+\rightarrow W^+b\bar{b}\) (right) in the same scenario as in Fig. 3. The orange curve corresponds to the full three-body width, while the contribution from on-shell intermediate particles (in particular \(t\rightarrow W^+b\)) is shown in green. The resulting genuine three-body contribution (beyond the two-body approximation) is shown in red and corresponds to the piece dominated by Sudakov double logarithms. We observe that this off-shell component (we continue to use this terminology below) is sizable and of competitive magnitude as compared to the two-body widths.

In the lower row of Fig. 4, we display the off-shell component of the three-body widths for the various relevant three-body channels (restricting ourselves to top and bottom quarks in the final state). Their sum is displayed in red. We observe that these contributions roughly reach the magnitude of the electroweak effects in the two-body width, corresponding to the difference between the blue and red curves in Fig. 3. We also stress that the three-body contribution is not of negligible size as compared to the two-body widths, so that we expect a sizable impact at the level of the branching ratios.

From the IR nature of the leading off-shell effects, one can expect the (off-shell) three-body widths to exhibit soft and collinear characteristics. We illustrate this fact in Fig. 5 for the channel \(h_2\rightarrow W^-t\bar{b}\), with the three cases \(M_{h_2}\approx 1\) TeV (green), 10 TeV (orange) and 100 TeV (red). The differential cross-section is shown in the upper row, in terms of the energy of the *W* boson \(E_W\) (left), and in terms of the angle between the *W* and the *b* antiquark \(\theta _{Wb}\) (right), in the rest-frame of the Higgs boson. We observe that the differential cross-section peaks at lower values of \(E_W\sim 100\text {--}200\) GeV as well as at \(\theta _{Wb}\sim 0\) (collinear *W* and \(\bar{b}\)) or \(\pi \) (collinear *W* and *t*, or collinear *t* and \(\bar{b}\)). The visible spikes in the upper-left plot slightly above \(E_W\sim 0.1,1,10\) TeV for \(M_{H^\pm }=1,10,100\) TeV are associated to the threshold for the on-shell decay \(h_2\rightarrow t\left( \bar{t}\rightarrow W\bar{b}\right) \); the corresponding width is subtracted from the three-body width as a Breit–Wigner distribution that, however, does not perfectly match the threshold effect.

In the lower row of Fig. 5, we show how much of the off-shell three-body decay width can be actually observed when requiring a minimal energy of the radiated *W* (left), or when requiring angular separations of \(\theta _0\in \left[ 0,\,\pi /2\right] \) between all the particles in the final state (in the rest-frame of the decaying Higgs; plot on the right): while the width depends only weakly on the energy cutoff as long as the initial state is very heavy, up to \(40\%\) of the signal could be lost with an angular cut of \(20^{\circ }\approx 0.35\) for \(M_h\simeq 10\text {--}100\) TeV. Thus, these three-body decays may be experimentally challenging. In particular, the part of the signal that produces collinear pairs with a *W* boson and a *b* quark could be misinterpreted as originating in an on-shell *t*. On the other hand, a collinear *t*–*W* pair is unlikely to be mistaken for an on-shell *b*. The exact boundary between two-body and three-body decays on the experimental side would probably require a dedicated analysis, which goes beyond the scope of this paper.

The impact of the resummation of double logarithms can be observed on Fig. 6. The green curve corresponds to the approximation of the inclusive three-body widths that can be inferred from the Sudakov double logarithms of the two-body channels \(h_2\rightarrow t\bar{t},b\bar{b},\tau ^+\tau ^-\) (left) and \(H^+\rightarrow t\bar{b},\tau ^+\nu _{\tau }\) (right). It is roughly comparable with the result obtained from a direct calculation of the three-body channels (orange curve). The red curve implements Eq. (2.13). That this curve is somewhat below the green and orange ones can be expected from its definition: if we expand the exponential term in Eq. (2.13), it is obvious that we are adding a negative contribution from the Sudakov resummation to the widths obtained through explicit calculation.

In these estimates of the three-body widths, we stress that (as our educated guess) the same QCD and QED correction factors as in the two-body widths are applied, in particular with QCD-running Yukawa couplings at the scale of the mass of the decaying Higgs. While these effects are formally of higher order, they have a sizable numerical impact on the widths. At \(M_{h_2}\approx 100\) TeV for example, the QCD running represents a correction factor of about 1 / 4, meaning that, evaluated with ‘pole’ Yukawa couplings, the three-body widths would dominate the two-body channels instead of being comparable. Yet, the QCD\(\times \)EW order is in principle not under control in our calculation, meaning that the large difference between the widths obtained from ‘pole’ and ‘running’ Yukawa couplings should be interpreted, strictly speaking, as a contribution to the theoretical uncertainty.

**Branching ratios** The Higgs branching ratios are often more helpful quantities than the decay widths, as they are more easily accessible experimentally. The three-body decays then affect all decay channels through their intervention in the full Higgs widths. We stress that the final states including the radiation of electroweak gauge bosons and Higgs bosons are clearly distinguishable from the two-body final states and that counting e. g. \(W^-t\bar{b}\) as a \(b\bar{b}\) event is only weakly motivated from an experimental perspective.^{3} We thus define the branching fractions as actual ratios of the two-body decay widths by the total width.

On the left-hand side of Fig. 7, in the upper row, we first display the total widths of the heavy \({\mathcal{CP}}\)-even state for the three considered approximations: the one-loop corrections to the two-body decays (orange curve) lead to a suppression of the total width (as compared to the effective tree-level order, in green), consistent with what was observed in Fig. 3, and mainly driven by the Sudakov double logarithms of Eq. (2.4). Nevertheless, the inclusion of the three-body decays (red) approximately restores the magnitude of the (QCD/QED-corrected) tree-level total width: this is an expected result, due to the cancellation of the double logarithms between the two- and three-body decays. On the right-hand side, we show the branching ratio \(\text {BR}{\left[ h_2\rightarrow b\bar{b}\,\right] }\), i. e. that of the naively dominant decay channel of the \({\mathcal{CP}}\)-even state. We observe that this quantity is suppressed, as compared to the tree-level prediction, by the decrease of the two-body width (orange line) but even more so by the inclusion of the three-body decays in the total width (red line). The impact of the one-loop corrections to the two-body decays is only at the percent level, while that of the inclusion of the three-body decays in the total width is of order \({\mathord {\sim }}\,10\%\) at \(M_{h_2}\approx 1\) TeV and reaches \({\mathord {\sim }}\,30\%\) at \(M_{h_2}\approx 100\) TeV. This can be easily understood when considering that the total width in the strict two-body approximation is suppressed in about the same proportion by the double logarithms as \(\Gamma \left[ h_2\rightarrow b\bar{b}\,\right] \) (its leading contribution): therefore, we expect little difference between the green and orange curves. On the contrary, when we include the three-body decays, the total width is approximately restored to its effective tree-level value, so that \(\text {BR}{\left[ h_2\rightarrow b\bar{b}\,\right] }\) becomes sensitive to the suppression of the associated decay width through double logarithms.

In the second row of Fig. 7, we turn to the leptonic decays \(h_2\rightarrow \tau ^+\tau ^-\) (left) and \(H^+\rightarrow \tau ^+\nu _{\tau }\) (right) as the typical search channels at proton–proton colliders. Contrarily to what we observed in the case of \(\text {BR}{\left[ h_2\rightarrow b\bar{b}\,\right] }\), the prediction for \(\text {BR}{\left[ h_2\rightarrow \tau ^+\tau ^-\right] }\) that is exclusively based on one-loop-corrected two-body widths (orange) is not suppressed, but enhanced. Indeed, in this approximation, the total width, dominated by \(h_2\rightarrow b\bar{b}\), receives a larger suppression as compared to the width of \(h_2\rightarrow \tau ^+\tau ^-\): the effect amounts to less than \({\mathord {\sim }}\,5\%\) depending on \(M_{h_2}\), and the green and orange curves show the same behavior – and so do they in the case \(H^+\rightarrow \tau ^+\nu _{\tau }\). However, the estimate at the full one-loop order (i. e. including the three-body decays; in red) shows the opposite trend. Again, as the three-body channels restore the magnitude of the total width to approximately its original effective value, there cannot be an associated enhancement any longer. On the contrary, the branching fractions become sensitive to the suppression of the tau-onic widths via Sudakov effects. Consequently, the discrepancy with the two-body approximation is sizable, loosely amounting to about \(10\%\) of the corresponding branching ratio at \(M_{h_2}\approx 1\) TeV and reaching \({\mathord {\sim }}\,30\%\) at \(M_{h_2}\approx 100\) TeV.

This short study shows that it is in fact misleading to assume that the one-loop corrections at the level of the two-body widths are sufficient to provide predictive branching ratios at the full one-loop order. We actually observe little difference with the effective tree-level in this approximation, while the inclusion of the three-body decays generates a sizable shift.

## 4 Conclusions

In this paper, we have emphasized the impact of the electroweak radiative corrections on the decays of heavy Higgs states. Focusing on the specific case of fermionic decays in the NMSSM, we have shown that such contributions are dominated by effects of IR type, namely Sudakov double logarithms, which could have a sizable impact on the magnitude of the two-body decay widths. This structure also provides a grasp on the electroweak corrections of higher order, since the leading double logarithms can be explicitly resummed.

In addition, we have stressed the relevance of three-body decay channels for a consistent evaluation of the total widths and branching ratios at the full one-loop order. The IR effects of the two-body widths are indeed mirrored by double logarithms in the radiation of electroweak gauge bosons, whereas the emission of light Higgs bosons only contributes at single-logarithmic order. Contrarily to the radiation of gluons and photons in QCD/QED, the radiation of electroweak gauge bosons leads to clearly distinguishable final states (due to the explicit IR cutoff fulfilled by the mass of the gauge bosons) and, in any case, the definition of an inclusive width is non-trivial (e. g. due to the flavor changes associated to emissions of *W* bosons). On the other hand, the off-shell three-body signals exhibit the expected soft and collinear characteristics that could make them difficult to extract experimentally.

Numerically, the inclusion of the three-body decays largely compensates the suppression of the two-body widths associated to electroweak effects, resulting in an approximately stable total width when simultaneously including or discarding the electroweak virtual corrections and real emissions. As a consequence, the branching ratios into two-body final states are fully sensitive to the suppression of the two-body widths associated to the double-logarithmic effects. We insist upon our conclusion that an estimation of the total Higgs widths and the branching ratios at the one-loop order solely from the loop-corrected two-body widths fails to capture the actual impact of the electroweak corrections.

In this work, we have focused on final states with SM fermions, since they are the expectedly leading decay channels of heavy doublet-like Higgs bosons involving SM particles in the final state – bosonic decays are suppressed in the decoupling limit. However, the main ingredients that we have discussed here also apply in alternative scenarios, e. g. decays into bosonic or SUSY final states. Indeed, three-body channels always need to be considered in parallel with one-loop corrections for a consistent order-counting and the Sudakov effects are always expected to emerge in electroweak corrections as long as there exists a hierarchy between the mass of the decaying Higgs (center-of-mass energy) and the decay products. However, these effects are only significant if the born-level amplitude itself is unsuppressed. By lowering the scale of the SUSY spectrum such that heavy Higgs bosons can decay into (comparatively) light pairs of neutralinos, charginos or sfermions, we should thus again recover sizable Sudakov corrections and relevant three-body decay widths in these channels with SUSY final states.

## Footnotes

- 1.
Decays of a heavy doublet into another heavy doublet state plus a gauge boson are kinematically inaccessible, in general. However, production of an off-shell gauge boson decaying into fermions has been considered e. g. in [66].

- 2.
Equivalently, it would be possible to factorize only the contributions to dimension 4 operators. For commodity in the processing of diagrams, it is more convenient to us to factor out the full SUSY contribution. The difference between the two procedures is a formally and numerically subleading term in the considered scenario.

- 3.
On the other hand, part of the signal \(h_2\rightarrow W^-t\bar{b}\,\) could be counted as \(h_2\rightarrow t\bar{t}\,\) events. However, for the considered value of \(\tan \beta \), most of the contributions from radiated gauge bosons are actually associated with the Yukawa coupling of the bottom and would thus be more appropriately attached to \(\Gamma \left[ h_2\rightarrow b\bar{b}\,\right] \) in an inclusive analysis.

## Notes

### Acknowledgements

We thank S. Heinemeyer, M. Stahlhofen and G. Weiglein for useful discussions. S. P. acknowledges support by the ANR Grant “HiggsAutomator” (ANR-15-CE31-0002).

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