Global fits in the Georgi-Machacek model

Off the beaten track of scalar singlet and doublet extensions of the Standard Model, triplets combine an interesting LHC phenomenology with an explanation for neutrino masses. The Georgi-Machacek model falls into this category, but it has never been fully explored in a global fit. We use the {\texttt{HEPfit}} package to combine recent experimental Higgs data with theoretical constraints and obtain strong limits on the mixing angles and mass differences between the heavy new scalars as well as their decay widths. We also find that the current signal strength measurements allow for a Higgs to vector boson coupling with an opposite sign to the Standard Model, but this possibility can be ruled out by the lack of direct evidence for heavy Higgs states. For these hypothetical particles, we identify the dominant decay channels and extract bounds on their branching ratios from the global fit, which can be used to single out the decay patterns relevant for the experimental searches.


I. INTRODUCTION
The discovery of a new scalar resonance at the LHC [1,2], consistent with the Higgs boson of the Standard Model (SM), confirms its particle content. Still several experimental observations, such as data on neutrino oscillations [3], beg for new physics explanations, whose effects are actively being looked for by the LHC experiments.
Among the well-motivated directions for new physics beyond the SM is the presence of an extended Higgs sector, which can lead to richer Higgs phenomenology at colliders. One possibility is the existence of additional Higgs triplet representations of SU (2), in which neutrino masses can arise from the interaction of the SM Higgs doublet with the triplet field, that acquires a vacuum expectation value (VEV) v ∆ after electroweak symmetry breakdown (EWSB). Particularly in order to avoid conflicts with the electroweak ρ parameter [3], the Georgi-Machacek (GM) model [4,5] adds one complex and one real scalar triplet in a way that ensures custodial SU (2) V symmetry is preserved in the scalar potential after the EWSB. The model predicts the existence of several Higgs multiplets, whose mass eigenstates form a quintet (H 5 ), one triplet (H 3 ) and two singlets (H 1 and h) under the custodial symmetry. In this work, we denote the 125-GeV Higgs boson by h.
The rich Higgs particle spectrum and associated attractive phenomenology deserve indepth studies, as it is of crucial importance to understand to which extent there is still room for new physics in the Higgs sector. Notably, if v ∆ is sufficiently large, we can have enhanced couplings between the SM-like Higgs boson and the weak gauge bosons. For example, κ W = 1.28 +0. 18 −017 is reported in a recent measurement by the CMS Collaboration [6], giving a hint for us to consider a Higgs sector with larger field representations [7,8]. The GM model serves as a minimal model with this feature. Modifications to the SM-like Higgs couplings with other particles can be probed by precise determination of the Higgs signal strengths at the LHC. Aside from loop-mediated processes, such data can constrain v ∆ and the mixing angle between the singlets α without the need to specify the heavy Higgs masses. In view of expected high precision in determining the Higgs couplings to other SM particles, Refs. [9,10] recently even computed the renormalized κ factors, defined to be Higgs couplings in the model normalized to their corresponding SM values, at the one-loop level. Since H 1 and h are related via an orthogonal rotation, these signal strengths also provide significant constraints on the couplings of H 1 to SM particles.
Earlier studies had shown various collider constraints on the parameter space of the GM model [8,[11][12][13][14][15][16][17]. In Ref. [15], for example, it was shown that after considering theoretical bounds (namely, the stability of the potential and perturbative unitarity at tree level), the LHC Higgs signal strengths, together with electroweak precision observables, a favored region in the (v ∆ , α) plane is chosen by the data.
In this work, we go beyond the existing literature by performing global parameter fits in the GM model, including up-to-date experimental results from Run 1 and Run 2 of the LHC, by making use of the HEPfit open-source package [18]. This approach is in stark contrast to studies that only examine specific benchmark scenarios (that may miss interesting possibilities), as all the model parameters are varied simultaneously in the fits and a model likelihood is obtained. The package also allows the possibility to identify which of the experimental data impose most stringent bounds. This paper is organized as follows. We start with a brief review of the GM model in Sec. II. Theoretical constraints on the scalar potential stability and perturbative unitarity at tree level are also included in our fits, as described in Sec. IV A. We then consider all available experimental data on Higgs boson signal strengths in Sec. IV B, including the γγ mode, thus extending the considerations in Ref. [15]. Constraints from sixty-seven heavy Higgs direct searches at the LHC as well as the basics of HEPfit package are described in Sec. IV C. They are included in our Bayesian analysis, and greatly extend the amount of constraints analyzed in previous works [12,14,15]. Combined results of the fits and discussions are presented in Sec. V. We close the paper with a summary of our findings in Sec. VI.

II. THE GEORGI-MACHACEK MODEL
In the GM model [4,5], SU (2)-triplet complex scalar χ and real scalar ξ are added to the SM particle content. Assuming that the custodial symmetry is preserved at tree level, we can write the SM doublet and new triplet scalar fields as a bi-doublet and a bi-triplet, respectively, After EWSB, the scalar fields have the VEV's given by Using, the above-defined fields, the scalar potential where σ a are the Pauli matrices, T a are the 3 × 3 matrix representation of the SU (2) generators, and the similarity transformation relating the SU (2) generators in the triplet and adjoint representations is given by Note that the triplet VEV is induced by the SM EWSB via the µ 1 interaction.
Under the custodial SU (2) V symmetry, the physical eigenstates can be written as a quintet H 5 = (H ++ 5 , H + 5 , H 0 5 , H − 5 , H −− 5 ) T with mass m 5 , a triplet H 3 = (H + 3 , H 0 3 , H − 3 ) T with mass m 3 and two scalar singlets H 1 and h, of which the former has the mass m 1 and the latter is identified with the 125-GeV scalar boson found at the LHC. The relations between the physical fields and the original fields can be found in, for example, Ref. [11]. Rotating from the original basis to the mass basis involves two mixing angles α and β, where α diagonalizes the singlet subspace and tan β ≡ v Φ /(2 √ 2v ∆ ) is used in the diagonalization of the Goldstone modes and the physical triplet states. In the limit of custodial symmetry, the states in each of the above-mentioned representations are degenerate in mass. An O(100) MeV mass splitting is expected among the states within the same representation because of custodial symmetry breaking by hypercharge interactions. In this article, we assume that h is the lightest scalar boson in the GM Higgs spectrum.
We list a few remarkable features of the GM model here. First, the hW W and hZZ couplings can be larger than the SM values at tree level. This does not happen in models extended with only singlet and/or doublet scalars. This feature is resistant to loop corrections, as explicitly shown in Refs. [9,10]

III. HEPFIT
The open-source package HEPfit is a multi-purpose tool to calculate many different highenergy physics observables and theory constraints in various models. It is interfaced with BAT [19] to perform Bayesian fits with Markov Chain Monte Carlo simulations. Here, we present the first results from the implementation of the GM model into HEPfit. The global fit allows us to scrutinize this model with unprecedented precision as it allows us to vary all GM parameters simultaneously, and thus guarantees that we do not miss important features when scanning over the parameter space. This method has also been used in the two-Higgs doublet model [20][21][22], and the GM implementation is partially based on the well-tested two-Higgs doublet model part of HEPfit in order to minimize possible sources of errors.
In our fits, we fix m h = 125.09 GeV and v = v 2 Φ + 8v 2 ∆ ≈ 246 GeV and all other SM parameters to their best-fit values [23]. We use the following prior ranges for the remaining GM parameters: where the masses m 1 , m 3 , m 5 of the H 1 , H 3 and H 5 bosons, respectively, are chosen to be heavier than the 125 GeV Higgs and lighter than 1 TeV, as we want to cover the ranges that are interesting for the LHC searches of heavy scalars. Accordingly, we also limit the absolute values of the trilinear couplings µ 1 and µ 2 to be below 1 TeV.
Concerning the heavy masses m 1 , m 3 and m 5 , our type of priors will depend on the set of constraints being used. For the direct searches, we will use flat mass priors, as the search limits depend on the masses linearly. As for the h signal strengths and the theory bounds, they depend on the squared masses. Therefore, we choose flat priors for m 2 1 , m 2 3 and m 2 5 between (150 GeV) 2 and (1000 GeV) 2 in this case. In the global fit to all constraints, we apply both types of priors in two separate fits and overlay both fits in the figures and for the extraction of the limits. (See also Appendix B of Ref. [21] for the same procedure in two-Higgs doublet model fits.)

IV. FIT CONSTRAINTS
In this section, we list the theoretical and experimental constraints imposed on the GM model parameter space in this analysis.

A. Theory constraints
We take into account two different sets of theoretical constraints: stability of the scalar potential and perturbative unitarity, both at tree level. Stability of the electroweak vacuum is imposed by requiring that the scalar potential be bounded from below, which places restrictions on the λ quartic couplings. We implement the constraints from Section 4 of Ref. [24].
Perturbative unitarity of the S-matrix of 2 scalars to 2 scalars scattering processes forces additional restrictions on the quartic couplings. We implement all seventeen constraints from the full S-matrix described in Ref. [25]. Here we take the stronger limits that the real part of the zeroth partial wave amplitude has an absolute value less than 1/2.
While the theory constraints are defined in terms of the quartic couplings of the scalar potential in Eq. (2), the following experimental bounds constrain the physical masses and the couplings of the scalars.

B. Higgs signal strengths
For the signal strengths computation, the predicted SM Higgs production cross-section σ and total decay width Γ are dressed with scale factors. For the production modes i = ggF, VBF, Wh, Zh, tth and the decay modes f = ZZ, W W, γγ, Zγ, τ τ, µµ, bb, we define r i and r f to be respectively the ratios of the production cross section σ i and the decay width Γ f with respect to their corresponding SM values. Therefore, the production cross section times the branching ratio for a particular channel in the GM model is given by with Γ SM and Γ GM being the total widths of the Higgs boson in the SM and the GM model, respectively.
To quantify the deviation of the GM model from the SM, the signal strength of a process µ f i = µ i · µ f with the production channel i and the decay of h to an f final state is then defined as Each signal strength is computed in the narrow-width approximation, and depends on the GM h couplings to all final states. The values for all couplings are cross-checked with the predictions in Ref. [26]. [60] In lines three to twelve, we give all LHC and Tevatron references of the used signal strengths, ordered by production mechanism and √ s. For the LHC, we indicate the share of Higgs production in pp collisions for each channel in the second column. The background colors of the table cells give an idea about how precise the strongest signal strength measurement in a particular category is at present: green cells contain results with an uncertainty of less than 0.5 on µ, yellow cells have an uncertainty between 0.5 and 1, and red entries have not been measured with a precision smaller than 1 (see the text for more details). On the decays to Zγ and µµ, we only have information for pp production and assume the SM composition in the second column for them.
The experimental input values of the Higgs signal strengths are the same as in Ref. [62]. Instead of all ∼130 numerical signal strength inputs, we show in Table I the current sensitivity of the individual channels, indicated by the background colors. The quantityσ is the ratio of the smallest uncertainty of all individual measurements in one table cell (σ min ) and the weight of the corresponding production mechanism (w). For instance, in Ref.
[38], we can find that µ τ τ = 1.11 +0. 34 −0.35 in their VBF category, so σ min = 0.34 here. Note that the categories do not consist of only one production mechanism, and thus the given value is no measurement of µ τ τ VBF . The admixture (weight) of VBF is only 57%, and soσ ≈ 0.6 in this case. We stress thatσ depends on the individual measurements and not on the combination. It is only intended to give the reader a rough estimate of the achieved precision in every channel, and should not be understood as a quantitative statement. In the last two columns, we use the 8-TeV data from Ref. [   and H 0 5 in the GM model. Table II shows all searches for a scalar resonance decaying into fermions or gauge bosons, and in Table III we list the cases with decays including one or two Higgs bosons. In Table IV, we list all searches for singly and doubly charged heavy scalars considered in our fits. Note that we are not sensitive in this model to the doubly charged Higgs searches in Refs. [88][89][90], where a 100% branching fraction to leptons is assumed and the decay of H ±± 5 to W ± W ± is suppressed, a scenario quite contrary to what we are considering here. The ATLAS searches for a doubly charged Higgs in Refs. [91,92] can have sensitivity in the two-lepton and three-lepton signal regions, and have been reinterpreted in the context of the Higgs triplet model [93] and GM model [14]. The limits provided by the experimental collaborations are not directly applicable to our case due to the B(H ±± → ± ± ) = 100% assumption, and, since in this work we are not formally recasting these searches, we choose not to include them in the fits.
The analyses in Tables II, III and IV provide either model-independent 95% confidence level upper limits on the production cross-section times branching ratios, σ · B, for different  production and decay modes, or they are quoted by σ · B/(σ · B) SM as a function of the resonance mass. If the experimental result includes the branching ratio into a specific final state in the upper limit, we spell out the channel, using parentheses to combine particles which stem from a primary decay product. Whenever a secondary final state is written inside square brackets, it means that we are quoting the limit on the primary final state measured through that particular secondary final state. In order to assess which parts of the GM model parameter space are favored after imposing these constraints, we first calculate the theoretical production cross-section times branching ratio, σ ·B, for all modes. For the neutral H 0 1 , H 0 3 and singly charged H ± 3 states, we calculate σ·B taking inputs from the two-Higgs doublet model already implemented in HEPfit [20,21], and rescale it to the GM model. We make use of the cross-section tables computed in Refs. [20,21] and calculate all branching ratios taking inputs from the couplings defined in the Appendix of Ref. [120]. For the quintet, we make use of the 13-and 8-TeV production cross-section tables from the LHC Higgs Cross-Section Working Group [121], including VBF production cross-sections for H ±± 5 , H ± 5 , and H 0 5 . The remaining VH modes and pair production of doubly charged H ±± 5 are calculated with MadGraph5_aMC@NLOv2.6.1 [122] at the leading order, taking the spectrum of the model generated with GMCalc [26] as input.
In order to compare the specific σ · B (calculated in each case as above) with the experimental upper limit, we define a ratio for the theoretical value and the observed limit, to which we assign a Gaussian likelihood with zero central value, which is in agreement with the null results in the searches of heavy scalars so far.

V. RESULTS
Here we show the impact of all the constraints considered on the GM model. We first discuss the effect of the measured h signal strengths. In Figure 1, we show the individual FIG. 1. Impacts of Higgs signal strengths on the v ∆ -α plane (left) and on the relative one-loop couplings of h to γγ and Zγ, r γγ and r Zγ , respectively (right). The 95% probability contours are shown from fits to the data for h decays to γγ (red), Zγ (yellow) , W W (blue), ZZ (green), bb (cyan), τ τ (purple) and µµ (orange). The combined fit to all h signal strengths is shown in grey.
impacts of specific decay categories on the α-v ∆ plane and on the plane of the effective loop couplings of h to γγ and Zγ, as well as the combination of all signal strengths. While the colored contours represent the allowed regions with 95% probability for each decay mode, the grey region gives the combined fit.
Two allowed grey regions can be seen in the left panel of Figure 1. The bigger region close to α ≈ 0 • (corresponding to the decoupling limit of the model) shows that v ∆ cannot exceed ≈ 50 GeV, and negative α is mostly favored. The other allowed solution close to α ≈ 61 • and v ∆ ≈ 77 GeV features a negative sign for the h couplings to vector bosons relative to the SM (r ZZ = r W W = −1). This region was not identified before as a viable possibility in the GM model (see, for instance, Ref. [15]), highlighting the advantages of using a global fitter 1 . The parameter space in the α-v ∆ plane is smaller in size as compared to the one given in Ref. [15], as now we see that α cannot reach beyond −30 • . This is due to the much larger data set on h signal strengths made available in the recent years as well as the addition of the γγ signal strengths.
In the right panel of Figure 1, we show the probability contours in the r Zγ -r γγ plane, illustrating the impact on the one-loop couplings of h to γγ and Zγ relative to the SM. The information on the loop couplings is complementary to the tree-level couplings, which can be purely determined for a given pair of α and v ∆ from the left panel. We observe a solution around the SM values, while a much smaller h coupling to Zγ than the SM also remain to be allowed, since so far we only have upper limits on the Zγ signal strength.
In Figure 2 we can see the effect of individual sets as well as all constraints in the v ∆ -α plane (top row), the α-m 1,3,5 plane (middle row), and v ∆ -m 1,3,5 plane (bottom row). After considering all constraints, the "wrong sign" region from Figure 1 gets excluded by the direct searches. The effect of the direct searches pushes v ∆ further down to less than about 25 GeV in the combined fit. Note that the decoupling limit [120] (α, v ∆ ≈ 0) is not favored in the fit due to the choice that our mass priors only go up to 1 TeV.
Looking at the plots in the bottom row, we note that the exclusion of the wrong sign limit comes from the H 5 searches. The effect comes mostly from the searches for the doubly charged H ±± 5 . The suppression of v ∆ stems from H 5 and H 3 searches, where mostly experimental limits on the neutral H 3 constrain v ∆ for m 3 between 200 and 400 GeV. In the middle row of Figure 2, a rather constant region of α close to 0 • is favored across the scanned mass range. Note the vertical green band in the α-m 1 plane around 250 GeV for direct searches is an effect of our implementation: for |α| > 12 • , the branching ratio of H 1 → hh is above 90%, but this can only be counterbalanced by the negative results of direct searches a few GeV above 250 GeV because of the finite step size in our interpolation. Figure 3 shows the effect of the theory bounds, h signal strengths, direct searches and all constraints on the mass differences. Again, the colored contours represent the allowed regions with 95% probability except for the theoretical constraints, for which we assume flat likelihoods. Hence, the 95% contours would only reflect on the priors, and the 100% contours are used for theory. Here we can see the power of the global fit. The individual sets of experimental constraints are not very strong in the m 5 − m 3 vs. m 5 − m 1 and m 5 − m 3 vs. m 3 − m 1 planes. The most dominant constraints come from the theoretical bounds, even though they still allow for a sizable region in the mass difference planes. However, once we combine the limits in the α-v ∆ plane from the LHC experiments with the theoretical conditions in the global fit, the region that survives at 95% shrinks to a thin strip for  After considering all the direct searches from the previous section, we find that the most powerful experimental analyses in constraining this model involve searches for the H 0 1 and H ±± 5 bosons. The effect on H 0 3 and H ± 3,5 is not as strong. In order to give a closer insight into our treatment of the direct searches and also to help the experimental collaborations better appreciate which search channels are more relevant or useful to the model, we show in Figure 4 four of the most constraining searches for heavy scalar resonances implemented in this work. These include the ATLAS searches for pp → H 0

13
. The grey regions in the background delimit the available GM space if we do not apply any constraint in the fit. We show the 100% prior ranges, but also the 95% prior regions, which in the H 1 → hh case differ by about one order of magnitude in σ · B. All these four searches cut away a sizable portion of the allowed parameter space, ranging from a difference of less than one order of magnitude between the H 5 → ZZ search limit and the grey contour to more than two orders of magnitude in σ · B for the searches of H 1 → hh and for the pair production of doubly charged scalars. Comparing this to the role of the individual searches in the global fit (yellow contour), we observe that the search in the left column is not quite relevant, while the channels in the right column yield the strongest constraint for m 1 between 500 and 800 GeV and for m 5 between 200 and 600 GeV, respectively.
We present in Table V the 95% probability ranges of the model parameters from our global fit. We do not get limits for the trilinear couplings µ 1 and µ 2 . The upper limit of 80 GeV on m 1 − m 3 enables us to exclude the decays H 1 → H 0,+ 3 H 0,− 3 , H 1 → H 3 Z as well as H 1 → H + 3 W − with a probability of 95%. Limits on derived quantities such as total decay widths and branching ratios for the H 1,3,5 scalars are presented in Table VI. For the total decay widths of the heavy singlet, triplet and quintet particles, we observe that they cannot exceed 90 GeV, 44 GeV and 11 GeV, respectively. For the SM-like Higgs boson, we obtain a probability range on Γ h between 3.9 and 4.5 MeV. These limits on the heavy Higgs decays serve as a guidance for the LHC experiments in the design of new searches for the scalars in the GM model.

VI. SUMMARY
We have performed global fits in the Georgi-Machacek model for the first time, making use of the HEPfit package and the latest experimental data. We consider constraints from both theory (stability of the scalar potential and perturbative unitarity) and LHC Higgs observables. These include several up-to-date experimental results from Run 1 and Run 2 of the LHC, including all the data on Higgs boson signal strengths and sixty-seven searches sensitive to the neutral, singly charged and doubly charged heavy Higgs particles of the Georgi-Machacek model. By considering only the signal strengths for the SM-like Higgs boson, we have found a previously unexplored region in the v ∆ -α plane, featuring a negative sign in the Higgs couplings to vector bosons with respect to the SM couplings. This solution around v ∆ ≈ 77 GeV and α ≈ 61 • cannot be ruled out by the signal strength data alone, but disappears as soon as direct search constraints are also imposed in the fit. The latter, especially the hunt for CP-even scalars, strongly constrains the vacuum expectation value of the Higgs triplet fields.
Combining the LHC bounds with the theory constraints in a global fit, we extract 95% probability limits on several Georgi-Machacek parameter regions and phenomenologically relevant quantities, which are significantly stronger than the bounds one would obtain when applying only one of the aforementioned sets of constraints. Among these are that α has to be between −24 • and 1 • and v ∆ smaller than 37 GeV. The latter means that cos β cannot exceed 0.43, which corresponds to an upper bound on sin θ H , where the mixing angle θ H is also used in the literature. We have found 95% limits on the differences between the heavy Higgs masses of values less than 350 GeV. The possibility of an H 1 decaying to H 3 can be excluded. We obtain upper 95% bounds on the total decay widths of the Higgs states and on many branching ratios. For instance, the H ±± 5 boson cannot decay into two H ± 3 bosons. The existence of a singly charged H ± 5 and doubly charged H ±± 5 scalars is a distinctive feature of the Georgi-Machacek model. Ongoing searches at the LHC (see Table IV) directly constrain v ∆ . Current searches for H ±± in constraining the Georgi-Machacek model in a global fit, which we did not consider in this work. This motivates flexibility in the definition of the experimental benchmarks in these searches to cases where the branching ratio of H ±± 5 to leptons is small, and its decay to vector bosons dominates.