Contents

July 11, 2019

http://cern.ch/gfitter

Update of the global electroweak fit and constraints
on two-Higgs-doublet models

The Gfitter Group
J. Haller, A. Hoecker, R. Kogler, K. Mönig, T. Peiffer, J. Stelzer

  • Institut für Experimentalphysik, Universität Hamburg, Germany
    CERN, Geneva, Switzerland
    DESY, Hamburg and Zeuthen, Germany
    II. Physikalisches Institut, Georg-August-Universität Göttingen, Germany

  • Abstract — We present an update of the global fit of the Standard Model electroweak sector to latest experimental results. We include new kinematic top quark and boson mass measurements from the LHC, a result from the Tevatron, and a new evaluation of the hadronic contribution to . We present tests of the internal consistency of the electroweak Standard Model and updated numerical predictions of key observables. The electroweak data combined with measurements of the Higgs boson coupling strengths and flavour physics observables are used to constrain parameters of two-Higgs-doublet models.

1 Introduction

Since the 1990’ies, electroweak precision data from LEP and SLD [1] were used together with accurate Standard Model (SM) calculations to predict parameters of the theory. A first impressive confirmation of the predictive power of global fits in high-energy physics (HEP) was the discovery of the top quark at the Tevatron [2, 3] in 1995, with a mass in agreement with the predictions from global fits. Knowledge of the top quark mass () made it possible to constrain the mass of the Higgs boson (). Increasing experimental and theoretical precision and the inclusion of constraints from direct Higgs boson searches from LEP and Tevatron narrowed the allowed mass range over time [4, 5, 6, 7, 8]. The discovery of the Higgs boson at the Large Hadron Collider (LHC) [9, 10] with a mass around GeV impressively confirmed the SM at the quantum level. The historical development of the constraints is illustrated in Figs. 1 and 2, where the predictions of, respectively, and as derived from various global fits and direct measurements [2, 3, 11, 12, 13, 14, 15, 16, 9, 10, 17, 18, 19, 20] are shown versus time.

With the measurement of the electroweak sector of the SM is overconstrained and the strength of global fits can be exploited to predict key observables such as the boson mass and the effective electroweak mixing angle, with a precision exceeding that of the direct measurements [21]. Since the last update of our fit [22] improved experimental results have become available that allow for more accurate tests of the internal consistency of the SM. Among these are the first determination of the boson mass at the LHC by the ATLAS collaboration [23], new combined results of the top quark mass by the LHC experiments [14, 20], a new combination of measurements of the effective leptonic electroweak mixing angles from the Tevatron experiments [24], a Higgs boson mass combination released by the ATLAS and CMS collaborations [17], and an updated value of the hadronic contribution to the running of the electromagnetic coupling strength at the boson mass [25]. In the first part of this paper we present an update of the electroweak fit including these new experimental results and up-to-date theoretical predictions.

While the Higgs boson measurements so far agree with a minimal scalar sector as implemented in the SM, the question remains whether a more complex scalar sector may be realised in nature, possibly featuring a variety of Higgs boson states. Two-Higgs-doublet models (2HDM) [26] are a popular SM extension in which an additional scalar doublet field with hypercharge is added to the SM leading to the existence of five physical Higgs boson states, , , , , and , where the neutral may be identified with the discovered 125GeV Higgs boson as is assumed in this paper. The scalar boson has CP-even quantum number, is a CP-odd pseudo-scalar, and and carry opposite electric charge but have identical mass. No experimental hint for additional scalar states has been observed so far in direct searches [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43]. In this situation global 2HDM fits, exploiting observables sensitive to these additional Higgs boson states via quantum corrections, can be used to constrain the allowed mass ranges and 2HDM mixing parameters. In the second part of this article such constraints are derived from a global fit using a combination of electroweak precision data, flavour physics observables, the anomalous magnetic moment of the muon, and measurements of the Higgs boson coupling strength to SM particles.

Figure 1: Prediction of the top quark mass versus year as obtained by various analysis groups using electroweak precision data (grey [5], light blue [4], green [6]). The bands indicate the 68% confidence level. The direct measurements after the top quark discovery are displayed by the data points (orange [2, 3, 11, 15, 16], red [13, 14, 20], black [12]).
Figure 2: Prediction of the Higgs boson mass versus year as obtained by various analysis groups using electroweak precision data (grey [5], light blue [4], dark blue [6]) and including direct search results (green [6]). The bands indicate the 68% confidence level. The direct measurements after the Higgs boson discovery are displayed by the red data points [9, 10, 17, 18, 19].

2 Update of the global electroweak fit

The updated global electroweak fit presented in this section uses the Gfitter framework. For a detailed discussion of the experimental data, the implementation of the theoretical predictions, and the statistical procedure employed by Gfitter we refer the reader to our previous publications [8, 44, 21, 22]. A detailed list of all the observables, their values and uncertainties used in the fit, is given in the first two columns of Table 1. The description below discusses recent changes in the input quantities and calculations.

2.1 Input measurements and theoretical predictions

The electroweak precision data measured at the pole and their correlations [1] as well as the width of the boson have not changed since our last analysis [22]. The update to the most recent world average values for the running and quark masses [45] has negligible impact on the fit result. This is also the case for the Run-1 LHC average of the Higgs boson mass,  [17], which we use now instead of a simple weighted average.111The Run-1 result on was confirmed by ATLAS and CMS measurements at  [18, 19].

New results are available for several observables with high sensitivity and potentially significant impact on the fit. We include new measurements of the boson and top quark masses as described in the following sections. For the first time we include as a separate fit input (assuming no correlation with other measurements) the latest combination of measurements of the effective leptonic electroweak mixing angle from the Tevatron experiments222The measurements of ATLAS ( [46]) and CMS ( [47]) are not included in the fit because of their presently insufficient precision and unknown correlations.,  [24], and we use an updated value for the five quark flavour hadronic contribution to the running of the electromagnetic coupling strength at ,  [25].

W boson mass

The ATLAS collaboration has recently released the first LHC measurement of the mass of the boson [23]. Analysing their dataset ATLAS measures . We include this result in the fit by combining it with the Tevatron ( [48]) and LEP combinations ( [49]) as follows.

Using information from Ref. [48] we estimate the composition of individual statistical, experimental systematic and modelling uncertainties in the combined Tevatron result by . All statistical and experimental systematic uncertainties are assumed to be uncorrelated among the three input results (ATLAS, Tevatron, LEP) as is the modelling uncertainty from LEP. The impact of the unknown correlation among the modelling uncertainties affecting the ATLAS and Tevatron measurements has been studied by varying its value between zero and one. For a large range of correlations we observe a stable average of , which we use in the fit.333A central value of is obtained for all possible values of the model correlation, except for coefficients exceeding 0.9 for which a value of is found. A combined uncertainty of is obtained for correlations between 0.4 and 0.9, while smaller and larger correlation values yield and , respectively. These values have been consistently calculated using the Best Linear Unbiased Estimate (BLUE) [50] and the least-squares averaging implemented in Gfitter [8].

Top quark mass

For lack of a recent world average, we attempt here for the purpose of the fit a conservative combination of the most precise kinematic measurements obtained at the LHC. We combine the averages from ATLAS ([20] and CMS ([14], which are based on 7 and 8 data. These averages include results from the dilepton [51, 52, 53], lepton+jets [13, 54] and fully hadronic [55] channels. Assuming the overlapping fraction of the systematic uncertainties to be fully correlated (which corresponds to a correlation coefficient of 72% between the two measurements) we obtain the combined value (-value of ), which we use as input in the fit.

The latest average from the D0 collaboration  [16] is barely compatible with the aforementioned average of the LHC measurements. A combination of the D0 average with the LHC average would result in -values between and , depending on the assumed correlation between the systematic uncertainties. The result from the CDF collaboration,  [56], agrees with the LHC average, with -values between and depending on the correlation.

As in our previous work [22] we assign an additional theoretical uncertainty of to the value of from hadron collider measurements due to the ambiguity in the kinematic top quark mass definition [57, 58, 59, 60, 61], the colour structure of the fragmentation process [62, 63], and the perturbative relation between pole and mass currently known to three-loop order [64, 65, 66].

Theoretical calculations

The theoretical higher-order calculations used in Gfitter have not changed since our last update [22], except for new bosonic two-loop corrections to the vertex [67].

For the effective weak mixing angle we use the parametrisations provided in [68, 69, 67], which include full two-loop electroweak [68, 69] and partial three-loop and four-loop QCD corrections [70, 71, 72, 73, 74, 75, 76, 77]. For bottom quarks, the calculations from Refs. [67, 78] are used. The new bosonic two-loop corrections are numerically small. They shift the prediction of the forward-backward asymmetry for quarks by , which is two orders of magnitude smaller than the experimental uncertainty and thus does not alter the fit results. We use the parametrisation of the full two-loop result [79] for predicting the mass of the boson, where we also include four-loop QCD corrections [75, 76, 77]. Full fermionic two-loop corrections for the partial widths and branching ratios of the boson and the hadronic peak cross section are used [80, 81, 82]. The dominant contributions from final-state QED and QCD radiation are included in the calculations [83, 84, 85, 86, 87, 88]. The width of the boson is known up to one electroweak loop order, where we use the parametrisation given in Ref. [89].

The size and treatment of theoretical uncertainties are unchanged with respect to our last analysis [22].

2.2 Results

Free Fit w/o exp. input Fit w/o exp. input
Parameter Input value in fit Fit Result in line in line, no theo. unc.
[GeV] yes
[GeV]
[GeV]
[GeV] yes
[GeV]
[nb]
[GeV] yes
[GeV] yes
[GeV] yes
yes
yes

Average of LEP () and SLD () measurements, used as two measurements in the fit. The fit without the LEP (SLD) measurement gives ( ). Combination of experimental (0.46 GeV) and theory uncertainty (0.5 GeV).In units of . Rescaled due to dependency.

Table 1: Input values and fit results for the observables used in the global electroweak fit. The first and second columns list respectively the observables/parameters used in the fit, and their experimental values or phenomenological estimates (see text for references). The third column indicates whether a parameter is floating in the fit. The fourth column gives the results of the fit including all experimental data. In the fifth column, the fit results are given without using the corresponding experimental or phenomenological estimate in the given row (indirect determination). The last column shows for illustration the result using the same fit setup as in the fifth column, but ignoring all theoretical uncertainties.

The fit uses as input observables the quantities and values given in the left rows of Table 1. The fit parameters are , , , , , , , as well as ten theoretical uncertainty (nuisance) parameters constrained by Gaussian functions (see Ref. [22] for more details).

The fit results in a minimum value of for degrees of freedom, corresponding to a -value of . The results of the full fit for each observable are given in the fourth column of Table 1, together with the uncertainties estimated from their profiles. The fifth column in Table 1 gives the results obtained without using the experimental measurement corresponding to that row in the fit (indirect determination of the observable). The last column in Table 1 corresponds to the fits of the previous column but ignoring all theoretical uncertainties [22].

Figure 3: Left: comparison of the fit results with the input measurements in units of the experimental uncertainties. Right: comparison of the fit results and the input measurements with the indirect determinations in units of the total uncertainties. Analog results for the indirect determinations illustrate the impact of their uncertainties on the total uncertainties. The indirect determination of an observable corresponds to a fit without using the constraint from the corresponding input measurement.

The left-hand panel of Fig. 3 displays the pulls each given by the difference of the global fit result of an observable (fourth column of Table 1) and the corresponding input measurement (second column of Table 1) in units of the measurement uncertainty. The right-hand panel of Fig. 3 shows the difference between the global fit result (fourth column of Table 1) as well as the input measurements (first column of Table 1) with the indirect determination (fifth column of Table 1) for each observable in units of the total uncertainty obtained by adding in quadrature the uncertainties of the indirect determination and the input measurement. The analog result using the value of the indirect determination, trivially centered around zero, are shown to illustrate the impact of its uncertainty on the total uncertainty. As in our previous fits, a tension is observed in the leptonic and hadronic asymmetry observables, which is largest in the forward-backward asymmetry of the quarks, . The impact of the new Tevatron measurement on the fit result is small due to yet insufficient precision.

Figure 4: Comparison of the constraints on obtained indirectly from individual observables with the fit result and the direct LHC measurement. For the indirect determinations among the four observables providing the strongest constraints (namely , , and ) only the one indicated in a given row of the plot is included in the fit. The results shown are not fully independent.

Figure 4 displays the indirect determination of the Higgs boson mass from fits in which among the four observables providing the strongest constraints (namely , , and ) only the one indicated in a given row of the plot is included. The results are compared to the direct measurement as well as to the result of a fit including all data except the direct measurement. This latter fit gives the indirect determination

(0)

which is in agreement with the direct measurement within standard deviations. The value is lower by than in our previous result ([22] due to the lower value of used here. The reduced uncertainty of compared to previously, is due to the smaller uncertainty in . When assuming perfect knowledge of , and , the uncertainty is reduced by , and , respectively. The predictions of using , and (LEP and Tevatron) concur with earlier findings [8]. The predictions derived from the ATLAS and Tevatron measurements are in agreement with the direct measurement.

An important consistency test of the SM is the simultaneous indirect determination of and . A scan of the confidence level (CL) profile of versus is shown in Fig. 5 for the scenarios where the direct measurement is included in the fit (blue) or not (grey). Both contours agree with the direct measurements (green bands and ellipse for two degrees of freedom).

Figure 5: Contours at 68% and 95% CL obtained from scans of versus for the fit including (blue) and excluding the measurement (grey), as compared to the direct measurements (green vertical and horizontal bands, and two-dimensional and ellipses). The direct measurements of and are excluded from the fits.

Figure 6 displays fit profiles for the indirect determination of some of the electroweak observables.444The indirect determination profiles are obtained by excluding the input measurement of the respective observable from the fit (see figure legends). The results are shown for fits including (blue) and excluding (grey) the direct measurement highlighting the strong impact of the measurement on the fit constraints. The direct measurement of each observable with its uncertainty are indicated by the data points at . The detailed predictions of the fit are given in Table 1.

Figure 6: Scans of as a function of (top left), (top right), (middle left), (middle right), (bottom left) and (bottom right), under varying conditions. The results of the fits without and with the measurement of as input are shown in grey and blue colours, respectively. The solid and dotted lines represent the results when including or excluding the theoretical uncertainties. The data points with uncertainty bars indicate the direct measurements of a given observable.

The fit indirectly determines the mass to be

(0)

and the effective leptonic weak mixing angle as

(0)

When evaluating through the parametric formula from Ref. [69], an upward shift of with respect to the fit result is observed, mostly due to the inclusion of in the fit. Using the parametric formula the total uncertainty is larger by , as the global fit exploits the additional constraint from . The fit also constrains the nuisance parameter associated with the theoretical uncertainty in the calculation of , resulting in a reduced theoretical uncertainty of compared to the input uncertainty.

The mass of the top quark is indirectly determined to be

(0)

with a theoretical uncertainty of 0.6 induced by the theoretical uncertainty on the prediction of . The largest potential to improve the precision of the indirect determination of is through a more precise measurement of . Perfect knowledge of would result in an uncertainty on of 0.9.

The strong coupling strength at the -boson mass scale is determined to be

(0)

which corresponds to a determination at full next-to-next-to leading order (NNLO) for electroweak and strong contributions, and partial strong next-to-NNLO (NNNLO) corrections. The theory uncertainty of this result is , which is shared in equal parts between missing higher orders in the calculations of the radiator functions and the partial widths of the boson. The most important constraints on come from the measurements of , and , also shown in Fig. 6. The values of obtained from the individual measurements are (), () and (). A fit to all three measurements results in a value of , which is only slightly less precise than the result of the full fit. The results obtained for are stable with respect to additional invisible beyond-the-standard-model contributions to .

No significant deviation from the direct measurements is observed in any of these predictions. The indirect determinations of and outperform the direct measurements in precision while the indirect determinations of and are competitive to other experimental results.

Oblique parameters

Using the updated SM reference values GeV and GeV we obtain for the oblique parameters denoted , ,  [90, 91] the following values:

(0)

with correlation coefficients of between and , () between and ( and ). Fixing one obtains and , with a correlation coefficient of . The constraints on and for a fixed value of are shown in Fig. 7.

Figure 7: Constraints in the oblique parameters and , with the parameter fixed to zero, using all observables (blue). Individual constraints are shown from the asymmetry and direct measurements (yellow), the partial and total widths (green) and mass and width (red), with confidence levels drawn for one degree of freedom.

3 Global fits in the two-Higgs-doublet model

Combining information from the electroweak precision data, Higgs boson coupling measurements, flavour observables and the anomalous magnetic moment of the muon we derive in this section constraints on parameters of various 2HDM scenarios.

Besides the four mass parameters for the scalars, , , , and , the 2HDM introduces the angle , which describes the mixing of the two neutral Higgs fields and , and the angle that fixes the ratio of the vacuum expectation values of the two Higgs doublets, . We only consider 2HDM scenarios with a symmetric potential with a dimension-two softly broken term proportional to the scale parameter .

Depending on the Yukawa couplings of the two Higgs doublets, the 2HDM may introduce dangerous flavour-changing neutral currents (FCNCs) and CP violating interactions. CP conservation can be maintained by fixing the Higgs boson couplings for up-type quarks, down-type quarks, and leptons to specific values [26, 92]. In this work, four CP conserving 2HDM scenarios are studied. In the Type-I scenario, only one of the two Higgs doublets is allowed to couple to fermions, while the other couples to the gauge bosons. The Type-II scenario is defined by a separation of the Yukawa interactions: one Higgs doublet couples only to up-type quarks and the other only to down-type quarks and charged leptons. The Type-II 2HDM resembles the Higgs sector in the Minimal Supersymmetric Standard Model. The third, lepton specific scenario is similar to the Type-I model with the difference that leptons only couple to the other Higgs doublet that does not interact with the quarks. Finally, the fourth, flipped scenario is the same as the Type-II model with swapped lepton couplings to the Higgs doublets.

Throughout this section the lightest scalar Higgs boson, , is identified with the observed Higgs boson with mass fixed to  GeV [17]. If not stated otherwise, all other 2HDM model parameters are allowed to vary within the intervals: , , , , and . No contribution from new physics other than the 2HDM is assumed.

Direct searches for additional Higgs bosons in collider experiments can be interpreted in the context of the 2HDM (see, for example, Ref. [93]). However, due to the large freedom in the choice of the 2HDM parameters, these search results provide only weak absolute exclusion limits on the masses of the scalars. From searches for a charged Higgs boson by the LEP experiments [27] a lower limit of was reported for the Type-I scenario, while a limit of can be derived from searches at the LHC for the Type-II scenario [93]. Stronger mass limits mainly on can be obtained for specific regions of .

3.1 Constraints from Higgs boson coupling measurements

A second Higgs doublet modifies the coupling strengths of the lightest neutral Higgs boson to SM particles compared to those of the SM Higgs boson. The modifications depend on the 2HDM scenario and parameters in particular the angles and . Constraints on are derived from the joint ATLAS and CMS Higgs boson coupling analysis [94] in which measurements sensitive to five Higgs boson production modes (ggF, VBF, , , ) and five decay modes (, , , , ) were combined. We make use of the relative signal strengths defined as the ratio of measured over predicted cross section times branching ratio, . We include the 20 (out of the 25 possible) parameters determined by ATLAS and CMS together with their uncertainties and correlations. A validation of our results is discussed in the Appendix on page 4.

The corresponding SM predictions and uncertainties are taken from Ref. [95]. The signal strength measurements are compared with the theory predictions for the 2HDM calculated with the program 2HDMC [96].5552HDMC computes the couplings of all five Higgs boson states to SM particles for a given set of parameters in a CP conserving 2HDM with general Yukawa structure. From these couplings, production and decay rates of the Higgs boson states can be derived. Most decay widths are calculated at leading QCD order in 2HDMC. Higher order QCD corrections are included for couplings to fermion and gluon pairs. In the calculation of the for the 2HDM also the denominator is determined using 2HDMC for consistency. Since more precise theory predictions for the SM cross sections and branching ratios exist and are used for the normalisation of the results in [94], theory uncertainties in the SM prediction from [95] are taken into account as additional scaling (nuisance) parameters of the .

The constraints from the Higgs boson signal strength measurements on the four 2HDM scenarios are shown as 68% and 95% CL allowed regions in the versus plane in Fig. 8.666Theoretical bounds from positivity of the Higgs potential, tree-level unitarity, and perturbativity of the quartic Higgs boson couplings as implemented in 2HDMC were found to give no additional constraints in these figures.

The angles and are highly constrained in all 2HDM scenarios except for Type-I. The allowed parameter regions are concentrated in two bands corresponding to solutions with . For , the Yukawa structure of the SM is reproduced (alignment limit). The case differs from the SM-like Yukawa couplings by a sign flip that is still allowed by the combined coupling strengths measurements. These constraints are differently pronounced in the four 2HDM scenarios as they depend on the Yukawa coupling strengths. In the Type-I scenario (top left panel in Fig. 8) the Yukawa couplings of to all fermions are proportional to . The constraints are stronger in the other three scenarios as the Yukawa coupling for at least one fermion type is proportional to . In the flipped scenario (bottom right panel) only the Yukawa coupling to down-type quarks is given by , which is constrained by the measurements of . Measurements of give stronger bounds in the Type-II (top right panel) and lepton specific (bottom left panel) scenarios where the Yukawa couplings to leptons is given by . In all scenarios, the measurements of Higgs boson decays to and boson pairs disfavour large values of . Similar constraints have been obtained by the ATLAS collaboration [97].

Figure 8: Results from 2HDM fits using the ATLAS and CMS combined Higgs coupling strength measurements. Shown are allowed parameter regions (68% and 95% CL) for the four 2HDM scenarios from scans of versus : Type-I (top left), Type-II (top right), lepton specific (bottom left) and flipped (bottom right) 2HDMs. The figure insets show a zoom of the region with .

3.2 Constraints from flavour observables

Because tree-level FCNC transitions are forbidden by construction in the four 2HDM scenarios considered, flavour violation only arises at loop level by the exchange of a charged Higgs boson with observable strength depending on the parameters and .

Experimental input data and theory calculation

Observable Value Reference
for  GeV [98, 99, 100]
 [98, 101]
[98, 102]
[98]
(CMS) [103]
(LHCb) [104]
(CMS)  [103]
(LHCb)  [104]
[98]
[98]
 ps [98]
 ps [98]
[45]
Table 2: Flavour physics observables and values used in the 2HDM fit.

The flavour physics observables taken into account in our analysis are listed in Table 2 and briefly described below.

For the branching fraction of the radiative decay with  GeV we use the value of the Heavy Flavour Averaging Group (HFLAV) [98] which combines measurements from the BABAR [105, 106, 107], Belle [108, 109, 110], and CLEO [111] experiments. The prediction for has been adopted from Ref. [100] and includes QCD corrections up to NNLO [112]. We make use of the code implementation kindly provided by M. Misiak.

HFLAV also combined measurements of the semileptonic decay ratios of neutral mesons by BABAR [113, 114], Belle [115, 116, 117], and LHCb [118] with a correlation of between the two observables that is taken into account in the fit. The prediction of  [101, 102, 119] includes tree-level contributions of a charged Higgs boson and is based on form factors evaluated in Heavy-Quark Effective Theory. Variations of the parameters , , , and are included in the fit with values and correlations taken from Ref. [98].

For the branching ratio we use the HFLAV average [98] of measurements from BABAR [120] and Belle [121]. For the prediction of in the 2HDM we use the calculation from Ref. [122], which contains tree-level contributions of a charged Higgs boson where the leading corrections are resummed to all orders [122]. The theoretical uncertainties in and (see below) are included.

The latest measurements of and from LHCb  [104] are combined in our fits with the CMS result [103], assuming them uncorrelated. Their theoretical predictions in the 2HDM include NLO corrections given in Refs. [123, 124]. The SM contribution to these observables are known up to three-loop level in QCD and include NLO electroweak corrections [125, 126, 127]. The predictions depend on the CKM matrix elements and or , respectively, and on the respective hadronic parameters and . Uncertainties in these parameters are taken into account in the fit.

The charged Higgs boson of the 2HDM contributes to the leptonic decays of mesons. For the observables and we use the HFLAV averages [98] of measurements from BABAR [128], Belle [129], and CLEO [130, 131, 132]. For the 2HDM predictions we use the analytic expression for the 2HDM tree-level contribution to from Ref. [133] that allows us to vary the dependencies on and in the fit.

The charged Higgs boson also contributes via box diagrams to the mixing of the neutral and mesons altering the mixing frequencies and/or . We use again the HFLAV [98] experimental averages for these quantities. Their predictions in the 2HDM are obtained from analytic expressions of the full one-loop calculation of Refs. [134, 119] neglecting small terms proportional to . The predictions depend on the CKM matrix elements and , the bag parameters and , and the decay constants and , respectively, and the correction factor .

Finally, the 2HDM contributes at leading order to the ratio for which we use a value adopted from Ref. [45], based on the measurement of the kaon decay rates [135], and the 2HDM prediction from Ref. [119]. The ratio involves the CKM matrix elements and , the decay constants and , and an electromagnetic correction .

As input values for the unitarity CKM matrix we use the latest available results for the all-orders Wolfenstein parameters , , , from Ref. [136, 137, 138], taking them uncorrelated. A fully consistent analysis would require a combined fit of the Wolfenstein and 2HDM parameters within the 2HDM [139], which is however beyond the scope of this paper. Studies in Ref. [119] and by ourselves have shown that the numerical impact of the 2HDM on the CKM parameters is modest. For the CKM element , occurring mainly in the prediction of the leptonic branching fraction, we take the average of inclusive and exclusive measurements [140] instead of the CKM fit prediction to allow for a more conservative uncertainty in view of the tension between the inclusive and exclusive results.

The input parameters used in the fit are summarised in Table 3.

Parameter Value Parameter Value
 MeV
 MeV
Table 3: Parameters used in the fit to the flavour observables. Most values are taken from latest available version of the CKM fit [138]. For the CKM matrix element we use the average of inclusive and exclusive measurements [140], while all other CKM matrix elements are calculated from the Wolfenstein parameters. The parameters related to the measurements, , , , are taken from Ref. [98]. Value and uncertainty for are taken from Ref. [141].

Results

Since most flavour observables are only sensitive to and , separate scans of these parameters are performed for each observable. The other 2HDM parameters are ignored in these scans, with the exception of , where in addition , , and are allowed to float freely within the bounds defined in the introduction of Section 3 as these two observables depend at NLO level on these parameters. In all fits the CKM matrix elements and the other parameters given in Table 3 are allowed to vary within their uncertainties.

Figure 9: Excluded parameter regions (95% CL) in the versus plane from individual observables for the four 2HDM scenarios considered: Type-I (top left), Type-II (top right), lepton specific (bottom left), flipped (bottom right).

Figure 9 shows for the four 2HDM scenarios the one-sided 95% CL excluded regions in the versus plane as obtained from fits using the most sensitive individual flavour observables. The CLs are derived assuming a Gaussian behaviour of the test statistic with one degree of freedom. The Type-I (top left) and lepton specific (bottom left) scenarios are only weakly constrained allowing to exclude . Stronger constraints are obtained for the Type-II (top right) and flipped (bottom right) scenarios in which in particular allows to exclude GeV.777Our results are compatible with those of Ref. [142], where limits on between 570 and 800 GeV are reported for the Type-II model, depending on the statistical method used (the CL has a relatively weak gradient versus and thus exhibits a strong numerical sensitivity to the details of the interpretation). Similar exclusion limits on can be achieved in a complex 2HDM (C2HDM), which features additional mixing between the neutral CP-even and CP-odd Higgs bosons [143].

The measurements of and differ from their SM predictions [101, 102, 144]. In the 2HDM only the Type-II scenario features a compatible parameter region (at large and relatively small , not shown in the upper right plot of Fig. 9), which is, however, excluded by several other observables. Similar results have been reported in Ref. [119]. Because of this incompatibility and are excluded from our analysis in the following.

3.3 Constraints from the anomalous magnetic moment of the muon

The measured value of the anomalous magnetic moment of the muon shows a long-standing tension with the SM prediction of  [145, 25], where the first uncertainty is due the the measurement and the second the prediction (see also the recent reanalysis in Ref. [146]). Loops involving 2HDM bosons can modify the coupling between photons and muons. We have adopted the two-loop 2HDM prediction of from Ref. [147], which depends on all 2HDM parameters. We make use of the code implementation kindly provided by H. Stöckinger-Kim.

Figure 10: 2HDM fits using the anomalous magnetic moment of the muon as input. Shown are allowed 68% and 95% CL regions in the versus plane for the four 2HDM scenarios considered: Type-I (top left), Type-II (top right), lepton specific (bottom left), and flipped (bottom right).

Figure 10 shows the 68% and 95% CL allowed regions in the versus plane for the four 2HDM scenarios using only as input. All other parameters of the 2HDM are left free to vary within their respective bounds. Compatibility is found in a narrow band with and below about 600 GeV (depending on the scenario), as well as for a region with larger that broadens with decreasing . When combined with the constraints from the other flavour observables (cf. Fig. 9), values of above about 510 remain allowed.

3.4 Constraints from electroweak precision data

The electroweak precision data can be used to constrain the 2HDM via the oblique parameters determined in Eq. (0). We use the predictions from Refs. [148, 149, 150] similar to our previous analysis [44]. The oblique corrections to electroweak observables in the 2HDM are independent of the Yukawa interactions and their impact is identical in the four 2HDM scenarios considered.

Figure 11 shows the 68% and 95% CL allowed parameter regions in the neutral Higgs-boson mass plane versus for fixed charged Higgs-boson masses of 250, 500, and 750 GeV as obtained from fits using only the oblique parameters as input. All other parameters of the 2HDM (including ) are free to vary in these scans. While no information on the absolute mass scale of the 2HDM bosons is obtained from the electroweak data, relative masses are constrained. In our previous analysis [44] we showed that the oblique parameters constrain the values of and to be close to for fixed . Removing this restriction (cf. Fig. 11) relaxes the constraint to having either close to , or larger than .

Figure 11: 2HDM fit results using the oblique , , parameters. Shown are allowed 68% and 95% CL regions in the versus plane for fixed benchmark values of . The constraints are independent of the 2HDM scenario.

3.5 Combined fit

We combine in this section the 2HDM constraints from the Higgs-boson coupling strength measurements, flavour observables, muon anomalous magnetic moment, and electroweak precision data.

Figure 12 shows for the four 2HDM scenarios considered the resulting 68% and 95 % CL allowed regions in the versus plane for fixed (benchmark) charged Higgs-boson masses of 250, 500, and 750 GeV. All other 2HDM parameters are allowed to vary freely within their bounds. Depending on the 2HDM scenario and , the minimum values found lie between 48 and 59 for (corresponding to -values between 25% and 68%).

The combined fit leads in all four 2HDM scenarios to a strong alignment of either the or the boson mass with that of the boson, owing to the constraint on from the Higgs coupling strength measurements (cf. Fig. 8) in addition to those from the electroweak precision data. In this sense, the fit resembles the result from our previous analysis [44], but replacing the fixed restriction of by the Higgs couplings strengths measurements.

The absolute mass limits on obtained from the flavour observables in the Type-II and flipped scenarios (cf. Fig. 9) exclude the low- benchmarks, as indicated by the hatched regions in the two right-hand panels of Fig. 12 (where in addition different statistical assumptions are compared: one-sided versus two-sided test statistic and one versus two degrees of freedom888The limits obtained for a two-sided test statistic and two degrees of freedom have been verified with a pseudo Monte Carlo study based on randomly drawn sets of the measurements used in the fit.). For these two scenarios pairs of (, ) masses below are excluded for any set of values of the other 2HDM parameters. For the Type-I and lepton specific scenarios no absolute limits on the Higgs boson masses can be derived.

Figure 12: 2HDM fit results using a combination of constraints from the Higgs-boson coupling strength measurements, flavour observables, muon anomalous magnetic moment, and electroweak precision data. Shown are allowed 68% and 95% CL regions in the versus plane for fixed benchmark values of and for the four 2HDM scenarios considered: Type-I (top left), Type-II (top right), lepton specific (bottom left), and flipped (bottom right).

4 Conclusion

We have presented results for an updated global fit of the electroweak sector of the Standard Model using latest experimental and theoretical input. We include new precise kinematic top quark and boson mass measurements from the LHC, a measurement from the Tevatron, and a new evaluation of the hadronic contribution to . The fit confirms the consistency of the Standard Model and slightly improves the precision of the indirect determination of key observables.

Using constraints from Higgs-boson coupling strength measurements, flavour observables, the muon anomalous magnetic moment, and electroweak precision data, we studied allowed and excluded parameter regions of four CP conserving two-Higgs-doublet models. Strong constraints on the extended Higgs boson masses are found for the so-called Type-II and flipped scenarios.

Acknowledgements

  • We are indebted to Mikolaj Misiak and Hyejung Stöckinger-Kim for providing the implementation of their calculations of and the muon anomalous magnetic moment in the two-Higgs-doublet models. We thank Rui Santos for helpful discussions and feedback on early stages of the paper. This work is supported by the German Research Foundation (DFG) in the Collaborative Research Centre (SFB) 676 “Particles, Strings and the Early Universe” located in Hamburg.

    Appendix

    To validate our implementation of the Higgs boson coupling measurements with respect to the full result from the ATLAS and CMS combination [94], we have performed a fit of a generic new physics parametrisation. Here, new physics effects are assumed to uniformly vary the coupling strength of the Higgs boson to vector bosons and fermions, respectively, according to linear modifiers and . No new particles are assumed to contribute to the Higgs boson production via loop diagrams and the branching fraction of the Higgs boson to unknown states is assumed to be zero. The constraints on and from the individual Higgs boson decay channels and their combination are shown in Fig. 13. We obtain the best fit values and with a correlation coefficient of . Decent agreement with Ref. [94] is seen.

    Figure 13: Validation of our implementation of the combined ATLAS and CMS Higgs boson coupling measurements: preferred regions from a two-dimensional scan of the coupling strength modifiers and for individual Higgs boson decay channels and their combination.

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