Uncovering the High Scale Higgs Singlet Model

The scalar singlet model extends the Standard Model with the addition of a new gauge singlet scalar. We re-examine the limits on the new scalar from oblique parameter fits and from a global fit to precision electroweak observables and present analytic expressions for our results. For the case when the new scalar is much heavier than the weak scale, we map the model onto the dimension-six Standard Model effective field theory (SMEFT) and review the allowed parameter space from unitarity considerations and from the requirement that the electroweak minimum be stable. A global fit to precision electroweak data, along with LHC observables, is used to constrain the parameters of the high scale singlet model and we determine the numerical effects of performing the matching at both tree level and 1-loop.


I. INTRODUCTION
The Higgs singlet model [1 -7] has been extensively studied as a simple extension of the Standard Model (SM) containing only one new particle. Depending on the potential parameters, the model can lead to a first order electroweak phase transition [8][9][10][11][12][13][14][15][16][17][18], making it highly motivated in addressing the problem of baryogenesis. It can also arise as the limiting case of many interesting models addressing the hierarchy problem [19,20] or even dark matter [21][22][23][24][25][26][27]. When the mass of the new scalar becomes much larger than the weak scale, the theory can be mapped onto an effective field theory. The utility and simplicity of the model thus makes it an ideal candidate for exploring the limits of an effective field theory framework in reproducing the features of the underlying UV models [28][29][30][31][32][33][34][35].
In the full UV complete singlet model, restrictions on the parameters can be found from fits to precision electroweak observables as well as LHC data. These limits can then be compared with limits found in the context of a low energy effective field theory. We consider an effective field theory in which the SM Higgs doublet is constrained to be an SU (2) doublet, the Standard Model effective field theory (SMEFT). At tree level, the singlet model generates only two SMEFT coefficients when matched at the UV scale [31,36]. The aim of this work is to examine to what extent the extraction of SMEFT coefficients from global fits at the weak scale gives information on the parameters of the UV complete singlet model [28,[32][33][34][37][38][39]. The focus is on understanding the numerical importance of various choices made when performing the low energy fits and to this end, we implement both tree and 1-loop matching [40][41][42] at the UV scale. We find that the effects of the 1-loop matching are typically rather small. Effects of O(10%) can be obtained only for rather large values of certain dimensionless parameters in the Lagrangian.
Section II contains a recap of the model and restrictions on the model parameters from unitarity and the minimization of the potential. Analytic results for electroweak precision observables in the singlet model are found in Section III along with a comparison between a global fit to electroweak precision observables (EWPOs) and a fit to the oblique parameters, and restrictions from unitarity and the minimization of the potential are in Section IV. The SMEFT matching with the singlet model at both tree and loop level is studied in Section V and a global fit to electroweak precision observables, Higgs, and di-boson data is presented.
Section VI has some conclusions.

II. BASICS
The singlet model we consider contains the SM Higgs doublet, Φ, and a scalar gauge singlet, S. The most general scalar potential is, The parameters can be redefined such that S ≡ x = 0. After spontaneous symmetry breaking, the 2 neutral scalars, Φ 0 and S, mix to form the physical scalars, h and H, with the physical masses, m h = 125.1 GeV and M H . The parameters of the model can be taken as, The other parameters of the Lagrangian are determined in the singlet model by: The Z 2 symmetric case has m ζ = t S = m ξ = 0 and x = 0.
The couplings of h to SM fermions and gauge bosons are suppressed relative to the SM Higgs couplings by a factor of cos θ, while the H couplings are suppressed by sin θ. We can thus immediately find a trivial limit on cos θ from Higgs production to SM particles X, (assuming no decays to invisible particles), 2 1 We note that for κv M , the mass of the new scalar, M H , comes from electroweak symmetry breaking and in this case the theory cannot be mapped onto the SMEFT [42,43]. Additionally, the kinematic distributions for hh production in this limit are quite different from those where M H primarily depends on M [44]. 2 If 2M H < m h then the decay h → HH is allowed, altering the limit on cos θ.
Naively combining the combined ATLAS results with 80 fb −1 [45] (6) we find at 95% C.L., For m h > 2M H , the naive limit of Eq. (7) does not apply because the h decays to HH must be included and this branching ratio is sensitive to the other parameters of the scalar potential. Limits on the singlet model from resonant double Higgs production are beginning to be competitive with those from single Higgs production for M H 700 GeV [47], although our primary focus here will be on M H ∼ (1 − 2) TeV.

III. RESTRICTIONS ON MODEL PARAMETERS
The parameters of the singlet model can be limited by a fit to the Z-and W -pole observables (we term this the EWPO fit): The SM results for these observables are well known [48,49]. In a previous study, Ref. [50], we computed the limits on the coefficients of an effective field theory that result from a fit to the observables of Eq. (8) computed to NLO in both QCD and electroweak interactions, and we apply an identical calculational framework here. The observables and SM theory numbers used in the current study can be found in in Table III of Ref. [50]. We take as our input parameters: G µ = 1.1663787(6) × 10 −5 GeV −2 , M Z = 91.1876 ± .0021 GeV, 1/α = 137.035999139(31), ∆α The one loop relation between the Fermi constant G µ and the vacuum expectation value v is, as usual, where, ∆r = ∆r SM + ∆r singlet .
We find the one-loop prediction for M W in the singlet model, This is in agreement with Ref. [52]. For a massless b quark, the total W decay width is, 3 The function A 0 is defined as where we calculate in d = 4 − 2 dimensions. B 0 is the Passarino-Veltman 2-point function, .
Analytic expressions for the remaining observables of Eq. (8) are given in the supplemental material attached to this note.
The finite b mass contribution to Z decays to bottom pairs is sensitive to the Higgs-b Yukawa coupling and generates non-oblique contributions. We compute R l , R b , R c and Γ Z for m b = 0 and find that the numerical effect is less than ∼ 2% for M H > 20 GeV, rising to ∼ 5% for M H ∼ 10 GeV, justifying the neglect of b mass effects in our fits.
We perform a fit, including correlations, to the observables of Eq. (8) to determine the maximum allowed value of sin θ for a given value of M H including all one-loop contributions.
It is of interest to compare the complete EWPO fit with the results using the oblique parameters only. Using the results of [53,54], we find that the differences between the Peskin-Takeuchi [55] variables in the Higgs singlet model and the SM take the form, where s 2 W = 1 − c 2 W = 1 − w/z is sin 2 θ W of the electroweak mixing angle and we define, We fit to the values in [56], with the correlation matrix, In Fig. 1  Eq. (7). The limits obtained from oblique parameters are in approximate agreement with those from the EWPO fit for a heavier second Higgs boson, although slightly different data sets and approximations were used in the numbers we fit to. 4 For the case where the second 4 We find rough agreement with Refs. [12,39] (the differences can be explained by the different numerical values of the input parameters) and disagree with the oblique parameter limits of Fig. 1 of [58]. We note that the curve labelled "Exact Singlet" on the RHS of Fig. 1 of Ref. [28] is the ST U result and has used a slightly different fit to the oblique parameters [59] from the PDG [56] results used here. The curve labelled Higgs in that plot is the prediction from fitting Higgs data within the context a SMEFT fit and thus differs from the SM Higgs coupling fit shown in Fig. 1.
Higgs is light, M H < m h , the limits obtained from the oblique parameters are not a good approximation of the complete EWPO fit.

IV. THEORETICAL CONSTRAINTS
In Section V, we will match the singlet model with a very heavy H to the SMEFT. Before we do so, we consider the theoretical restrictions on the singlet model parameters that are relevant for the matching.
A. Vacuum Structure of the Potential The first set of theoretical constraints on the singlet model come from requiring a suitable vacuum structure of the potential [5,7,10,60]. Demanding that the potential is stable at large field values leads to the requirement λ H , λ S > 0, and κ ≥ −2 √ λ H λ S [7], where λ H is determined by Eq. (4). Additional bounds result from requiring that the electroweak minimum be the global minimum of the potential. Following [7], we compute these bounds by finding all the extrema of the potential expanded around the electroweak vev as a function of (v, x), and then checking whether or not the value of the potential at (v = 246 GeV, x = 0) is the global minimum.
The extrema of the potential can be divided into two classes: extrema where v = 0, and those where v = 0. In the former case, the new extrema are denoted (v ± , x ± ) as in ref. [7], and tend to bound lower values of κ. In the latter case, the extrema are denoted by (0, x 0 ± ), and these tend to limit both large values of κ as well as large values of m ζ . An example of the vacuum structure is shown in Fig. 2, where we illustrate the regions excluded by the emergence of different global minima as well as the condition from vacuum stability. Fig. 3 illustrates how these bounds change as a function of the physical parameters. In particular,

A. One-loop Matching
When the mass of the heavy scalar is much larger than the weak scale and any relevant energy scales, the singlet model can be modeled by an effective field theory, with coefficients matched to the singlet model at the high scale, M . We retain only the dimension-6 operators, O (6) i ,and use the Warsaw basis [64] with the notation of Ref. [65].
The global fits of Ref. [28] were performed using tree level matching at the scale M . 5 It is of interest to implement the one-loop matching for the case of the singlet model and examine the numerical impacts. The coefficients at the matching scale, M , generically take the form, where c i (M ) is the tree level result and d i (M )/(16π 2 ) is the one-loop contribution at the matching scale. When the renormalization group evolution to the low scale µ R is included, In the case of the singlet model only two coefficients are generated at tree level [31,36,63,66], with all other c i (M ) = 0. However, there are many coefficients generated at one-loop at the matching scale, M [40][41][42]. The majority of these coefficients are proportional to the tree level coefficient, c H . We use the shorthand C Hu → C Hu , C Hc , C Ht , etc., and take y u = y c = 0, y t = M t √ 2/v (similarly we set all other y i = 0) and we further assume that C (1) Hq , C (1) Hl , and C

(3)
Hl are flavor diagonal and use an analogous shorthand. For convenience, we list the results of Ref. [40] in our notation: 6 The one-loop contribution d tH can be written in terms of c H and C H and is, where in the SMEFT the physical Higgs mass is determined in terms of the potential parameters to O(v 2 /M 2 ) by [65], and we define, where we note that Ref. [40] absorbs the factor of c H into the definition of λ used in the matching conditions, along with a relative factor of 2 in the definition of the quartic terms in the potential. 7 Eq. (31) represents the dimension-6 SMEFT limit of Eq. (4) for the relationship between the parameters of the potential and m h .
Finally, the coefficients generated at tree level also receive one-loop corrections, where, The terms of Eqs. (34) and (35) can be written in terms of c H , c H along with m ζ , λ S and κ, (m ξ can be written in terms of these parameters). The one-loop shift terms from canonically normalizing the Higgs kinetic energy are, The one-loop shift terms are O(v 4 /M 4 ) and can be neglected, since we consistently work to linear order in the coefficient functions.
After performing the one-loop matching at M , the renormalization group is used to evolve the coefficients to M Z , where the resulting coefficients can be compared with data. 8 The complete set of one-loop anomalous dimension matrices can be found in Refs. [67][68][69]. The inclusion of the one-loop matching makes a relatively minor difference in the evolution of 7 We drop the c HD term in Eq. (31) since it doesn't occur in the singlet model. 8 A more consistent approach would employ the 2-loop anomalous dimensions, however, these are not available for the SMEFT. C H and C H , as seen in Fig. 4 where we evolve from 2 TeV (note that M H is related to M by Eq. (4)). In Fig. 5, we show the effect of the one-loop matching on the evolution of C HD .
In this case, since C HD is zero at tree level, the contributions from the 1−loop matching and the renormalization group running are of the same order of magnitude and the effects are more significant. In Fig. 6 we show the relative size of the 1−loop matching compared to the tree level matching as the matching scale M is increased and the overall effects are between 10 − 30%. The size of the effects for C H and C H increase dramatically as the matching scale rises over a few TeV. This is due to the logarithmic running becoming large and in the case of C H , the 1-loop matching terms become of the same order as the tree level terms, implying that the perturbative expansion is no longer valid. are generated with a distinctive pattern. The 1-loop matching introduces a dependence on three additional parameter combinations beyond those at generated by the tree level matching and we take as our 5 unknown input parameters, M H , sin θ, m ζ , λ S , and κ. 9 The matching scale, M , is then calculated using Eq. (4). We match the SMEFT coefficients at M and use the 1-loop renormalization group equations to evolve the SMEFT coefficients to 9 The results used to include the effects of C H require |C H | (5 − 6) M/TeV 2 [70,71].
M Z where we fit to data.
The included data are identical to that of Ref. [28] and include Higgs coupling strengths from ATLAS [45], CMS Higgs coupling strengths [46], W + W − , W ± Z, W h and Zh differential measurements including QCD effects as in [72,73], and precision electroweak measurements including QCD and electroweak NLO effects from Table III of Ref. [50]. We determine the 95% confidence level limits using a χ 2 fit, including the new physics effects at linear order in the SMEFT coefficients. We separately show the limits from only EWPO limits in magenta and note that the 1-loop matching slightly improves the bound on sin θ.
Another interesting way to look at the results is to look at the maximum allowed value of sin θ as a function of the heavy Higgs mass, M H , for fixed values of κ, m ζ and λ S as shown in Fig. 8. We see that including the 1-loop matching changes the bound on sin θ only marginally. The effect is larger as κ is increased.
Single parameter fits to models with an additional scalar have been presented in Ref. [33] and updated in Ref. [34] using the dictionary of Ref. [66]. Assuming C H = −1/M 2 and C H = 0, they find a limit M H > 900 GeV at 2σ. Our tree level matching result of Ref. [28] is roughly compatible with this bound, although we find that the inclusion of the renormalization group running of the coefficients (in particular C HD which is generated by renormalization group running) is numerically significant, so the bounds cannot be directly compared. 10 10 Fits to the singlet model with 1-loop matching, but no renormalization group running, are given in Ref. [35], but the results are not in a form that we can compare with.

VI. CONCLUSIONS
We have re-examined the sensitivity of a global fit to electroweak precision observables and to Higgs and diboson data on the parameters of a scalar singlet model in both the full UV complete model and in the low energy approximation where the heavy scalar is integrated out and the parameters are matched to the dimension-6 SMEFT. In the full singlet model, we find equivalent limits on the allowed mixing angle from the complete EWPO fit and from the fit to the oblique parameters when M H is heavy. For the case with the second Higgs boson much lighter than M Z , the oblique parameter limits are not a good approximation to the full fit. When the new scalar is very heavy, we integrate it out and match to the dimension-6 SMEFT and then perform the global fit both using tree level and 1-loop matching at the high scale and derive limits on the parameters of the singlet theory from the SMEFT fit. We find that the effect on the fit of including the 1-loop matching is never larger than O(10%) and that the results are quite insensitive to variations in the singlet Lagrangian parameters other than the portal term, κ.