Higgs Boson Precision Analysis of the Full LHC Run 1 and Run 2 Data

We perform global fits of the Higgs boson couplings to the full Higgs datasets collected at the LHC with the integrated luminosities per experiment of approximately 5/fb at 7 TeV, 20/fb at 8 TeV, and up to 139/fb at 13 TeV. Our combined analysis based on the experimental signal strengths used in this work and the theoretical ones elaborated for our analysis reliably reproduce the results in the literature. We reveal that the LHC Higgs precision data are no longer best described by the SM Higgs boson taking account of extensive and comprehensive CP-conserving and CP-violating scenarios found in several well-motivated models beyond the SM. Especially, in most of the fits considered in this work, we observe that the best-fitted values of the normalized Yukawa couplings are about $2\sigma$ below the corresponding SM ones with the $1\sigma$ errors of 3%-5%. On the other hand, the gauge-Higgs couplings are consistent with the SM with the $1\sigma$ errors of 2%-3%. Incidentally, the reduced Yukawa couplings help to explain the excess of the $H\to Z\gamma$ signal strength of $2.2\pm 0.7$ recently reported by the ATLAS and CMS collaborations.


I. INTRODUCTION
Since the ATLAS and CMS collaborations have independently reported the observation of a new scalar particle in the search for the Standard Model (SM) Higgs boson in 2012 [1,2],1 more than 30 times larger number of Higgs bosons have been recorded by the both collaborations at the CERN Large Hadron Collider (LHC).Recently, ten years after the discovery, the two collaborations have presented the two legacy papers portraying the Higgs boson and revealing a detailed map of its interactions based on the precision Higgs data collected during the Run 2 data-taking period between 2015 and 2018 [3,4].During the Run 2 period at a center-of-mass energy of 13 TeV, the ATLAS and CMS collaborations have accumulated the integrated luminosities of 139/fb and 138/fb, respectively, which exceed those accumulated during the full Run 1 period by the factor of more than 5. 2Though there was a quite room for the new scalar boson weighing 125 GeV3 to be different from the SM Higgs boson around the discovery stage [6] but the increasing Higgs datasets soon revealed that they were best described by the SM Higgs boson [7].Around the end of the Run 2 period before the Run 2 Higgs data are fully analyzed, the five productions modes of gluon-gluon fusion (ggF), vector-boson fusion (VBF), the associated production with a V = W/Z boson (WH/ZH), and the associated production with a top-quark pair (ttH) had been extensively investigated and, impressively, the Higgs decays into a pair of b quarks [8,9] and a pair of τ leptons [10,11] were observed in single measurements leading to the firm establishment of third-generation Yukawa couplings together with the top-quark Yukawa coupling constrained by the ggF and ttH productions and the Higgs decay to two photons [12].Now, with the Run 2 Higgs data fully analyzed [3,4], the sixth Higgs production process associated with a single top quark (tH) starts to be involved and the Higgs boson decays into a pair of muons and Zγ are emerging.The direct searches for so-called invisible Higgs boson decays into non-SM particles have been also carried out.
In Refs.[3,4], the ATLAS and CMS collaborations scrutinize the interactions of the 125 GeV Higgs boson using the Higgs precision data recorded by their own experiments during the Run 2 period and independently yield the following Run 2 global signal strengths assuming that all the production and decay processes scale with the overall single signal strength: µ Global Run 2 ATLAS = 1.05 ± 0.06 ; µ Global Run 2 CMS = 1.002 ± 0.057 , in remarkable agreement with the SM expectation.In this work, by combining the ATLAS and CMS Run 2 data on the signal strengths as well as including, though statistically less important, the Run 1 LHC [13] and Tevatron global signal strengths, 4 we find the following global signal strength: µ Global 76 signal strengths = 1.012 ± 0.034 .
Upon the previous model-independent analyses [6,7,12], we have improved our analysis by including the tH production process to accommodate the LHC Run 2 data and by treating the ggF production signal strength beyond leading order in QCD to match the precision of the ever-increasing Higgs data now and after.
We demonstrate that our combined analysis based on the experimental signal strengths used in this work and the theoretical ones elaborated for our analysis reliably reproduce the fitting results presented in Ref. [13] (Run 1) and Refs.[3,4] (Run 2) within 0.5 standard deviations.Our extensive and comprehensive CP-conserving and CP-violating fits taking account of various scenarios found in several well-motivated models beyond the SM (BSM) reveal that the LHC Higgs precision data are no longer best described by the SM Higgs boson.Especially, in most of the fits, we observe that the best-fitted values of the Yukawa couplings are about 2σ below the corresponding SM ones with the 1σ errors of 3-5%.The reduced Yukawa couplings help to explain the combined H → Zγ signal strength of 2.2 ± 0.7 recently reported by the ATLAS and CMS collaborations [14].Accordingly, the SM points locate outside the 68% confidence level (CL) region mostly and even the 95% CL region sometimes in many of the two-parameter planes involved Yukawa couplings.We further note that the BSM models predicting the same scaling behavior of the Yukawa couplings to the up-and down-type quarks and charged leptons are preferred.On the other hand, the gauge-Higgs couplings are consistent with the SM with the 1σ errors of 2%-3%.Incidentally, we note that CP violation is largely unconstrained by the LHC Higgs data with the CL regions appearing as a circle or an ellipse or some overlapping of them in the CP-violating two-parameter planes.This paper is organized as follows.Section II is devoted to reviewing the ATLAS and CMS Run 2 data on the signal strengths as well as the Run 1 LHC and Tevatron ones used in this work.We compare our results on the global signal strengths and the signal strengths for the individual Higgs production processes and decay modes with those in    [3] used in this work (139/fb at 13 TeV).We refer to the website https://doi.org/10.17182/hepdata.130266for specific information.MH = 125.09GeV is taken.
We compare the ATLAS+CMS combined Run 1 data on signal strengths with the either ATLAS or CMS Run 2 data on signal strengths for various combinations of Higgs boson production and decay processes, see Table II and Tables III and IV.We note that the tH production process and the H → µµ decay mode have been newly measured and the mixedproduction modes such as ggF+bbH⊕VBF, WH⊕ZH, ttH⊕tH, ggF+bbH⊕ttH⊕tH, and VBF⊕WH⊕ZH are involved especially when the Higgs boson decays into a pair of fermions. 8Incidentally, in Run 2, the Higgs decay into ZZ * is measured also in the WH, ZH, and ttH production modes though the corresponding errors are still large.Further we observe that the ttH and tH production processes have been always combined except in the ATLAS measurement of the TABLE V. LHC signal strengths for individual Higgs production processes and decay modes.For the combined Run 1 signal strengths, see Tables 12 and 13 in Ref. [13].For the ATLAS Run 2 signal strengths, see Fig. 2 in Ref. [3] together with detailed information on them provided in the website https://doi.org/10.17182/hepdata.130266.The CMS Run 2 signal strengths are from Fig. 2 in Ref. [4].Note that the production signal strengths have been obtained assuming that the Higgs boson branching fractions are the same as in the SM and the decay signal strengths have been extracted assuming that the Higgs boson production cross sections are the same as in the SM.In Ref. [13] and Refs.[3,4] for the LHC Run 1 and Run 2 data, respectively, also presented are the individual signal strengths for Higgs boson production processes which have been obtained assuming that the Higgs boson branching fractions are the same as in the SM and the individual signal strengths for Higgs boson decay modes which have been extracted assuming that the Higgs boson production cross sections are the same as in the SM.We collect them in Table V. Comparing the ATLAS+CMS combined Run 1 signal strengths with the either ATLAS or CMS Run 2 ones, we observe that, overall, each of the signal strengths is more precisely measured with the errors reduced by the factor of about 2 and the most of their central values approach nearer to the SM value of 1. Especially, the error of the ttH production is reduced by the factor of about 3 and the production mode has been evidenced in Run 2. Looking into the ATLAS and CMS Run 2 signal strengths, we note that the ggF production signal strength µ ggF+bbH9 has been most precisely measured with the errors of 7%-8% followed by µ VBF with about 12% error.The signal strengths of the other production modes of WH, ZH, ttH(⊕tH) are measured with about 20% errors.For the decays, the γγ, ZZ * , W W * , and τ τ modes are measured with about 10% errors while the bb mode with 15 − 20% error.The µµ mode is slightly better measured by CMS with 45% error.

LHC
Lastly, also available are the global signal strengths which are simplest and most restrictive among all kinds of signal strengths since they have been yielded under the assumption that all the production and decay processes scale with the same single global signal strength independently of the production and decay processes: Run 1 = 1.09 +0.11 −0.10 [13]; µ Global Run 2 ATLAS = 1.05 ± 0.06 [3] , µ Global Run 2 CMS = 1.002 ± 0.057 [4] .
Together with the Tevatron global strength we observe that the global signal strengths have been always consistent with the SM value of 1 and are converging to the SM value of 1 with the ever-decreasing errors.Combining the three LHC global strengths and the Tevatron one, we have obtained For the above result, we assume that each global strength is Gaussian distributed and correlation among them could be ignored.
In this work, we use the following 76 Higgs signal strengths which have been extracted for different combinations of Higgs boson production and decay processes without imposing any assumptions: • Tevatron: 3 signal strengths based on 10/fb at 19.6 TeV, see Table I • LHC Run 1 (ATLAS+CMS): 20 signal strengths based on about 2 × 25/fb at 7⊕8 TeV, see which is consistent with the global signal strength given by Eq. ( 5) which is obtained from the four LHC and Tevatron global signal strengths.
Observing the difference between the two global signal strengths, see Eqs. ( 5) and ( 6), we also show the LHC signal strengths for individual Higgs production processes and decay modes in Table VI, obtained by using the LHC signal strengths in Table II (Run 1), Table III (Run 2 ATLAS), and Table IV (Run 2 CMS).For the production signal strengths, we neglect the tH production process in ATLAS Run 2 and all the mixed-production processes except ttH⊕tH.Note that, being different from the individual signal strengths in Table V, no assumptions have been imposed on the Higgs boson branching ratios and/or production cross sections.The combined individual production and decay signal strengths obtained by using all the 76 signal strengths are shown in the last line of Table VI.Since any information on correlations among Run 1, Run 2 ATLAS and Run 2 CMS datasets are not currently available, we ignore them accordingly.

III. FRAMEWORK
For the conventions and notations of the model-independent couplings of the 125 GeV Higgs-boson H to the SM particles, we closely follow Ref. [47].To be most general, we assume that H is a CP-mixed scalar.And then, in terms of the CP-violating (CPV) Higgs couplings, we calculate the theoretical signal strengths used for our global fits.Especially, we have included the production signal strength for the tH process to accommodate the new feature of the LHC Run 2 data and considered the ggF production process beyond leading order in QCD to match the level of precision of the LHC Run 2 data.

A. Higgs Couplings
The interactions of a generic neutral Higgs boson H with the SM charged leptons and quarks, without loss of generality, could be described by the following Lagrangian: where g S H f f and g P H f f stand for the H coupling to the scalar and pseudoscalar fermion bilinears, respectively, and they are normalized as g S H f f = 1 and g P H f f = 0 in the SM limit.The interactions of H with the massive vector bosons V = Z, W are described by in terms of the normalized couplings of g HW W and g HZZ with g = e/s W the SU(2) L gauge coupling and the weak mixing angle θ W : c W ≡ cos θ W and s W ≡ sin θ W .For the SM couplings, we have g HW W = g HZZ ≡ g HV V = 1, respecting the custodial symmetry between the W and Z bosons.
The loop-induced Higgs couplings to two photons are described through the following amplitude for the radiative decay process H → γγ: by introducing the scalar and pseudoscalar form factors denoted by S γ and P γ , respectively, with k 1,2 and ϵ 1,2 being the four-momenta and wave vectors of the two photons. 10Retaining only the dominant contributions from third-generation fermions and the charged gauge bosons W ± and introducing two residual form factors ∆S γ and ∆P γ to parametrize contributions from the triangle loops in which non-SM charged particles are running, the scalar and pseudoscalar form factors are given by where N f C = 3 for quarks and For the definitions and behavior of the loop functions F sf,pf,1 , see, for example, Ref. [47].Taking M H = 125 GeV, for example, one may obtain the following estimation of the form factors: in terms of the Higgs-fermion-fermion and HW W couplings given in Eqs. ( 7) and ( 8) supplemented by the two residual form factors ∆S γ and ∆P γ .In the SM limit where g HW W = g S H f f = 1 and g P H f f = ∆S γ = ∆P γ = 0, we have S γ SM = −6.542+ 0.046 i and P γ SM = 0.
The loop-induced Higgs couplings to two gluons are similarly described through the amplitude where a and b (a, b = 1 to 8) are indices of the eight generators in the SU(3) adjoint representation, k 1,2 the four momenta of the two gluons and ϵ 1,2 the wave vectors of the corresponding gluons.Referring to Ref. [47] again for the detailed description and evaluation of the amplitude, the scalar and pseudoscalar form factors are given by retaining only the dominant contributions from third-generation and charm quarks and introducing ∆S g and ∆P g to parametrize contributions from the triangle loops in which non-SM colored particles are running.Taking M H = 125 GeV, one might have in terms of the Higgs-fermion-fermion couplings given in Eq. ( 7) supplemented by the two residual form factors of ∆S g and ∆P g .We have S g SM = 0.636 + 0.071 i and P g SM = 0 in the SM limit.

B. Theoretical Signal Strengths
In this work, to calculate the theoretical signal strengths, we adopt the factorization assumption under which the production and decay processes are well separated like as in the resonant s-channel Higgs production in the the narrowwidth approximation and neglect nonresonant and interference effects.Under the assumption, for a specific productiontimes-decay process, each theoretical signal strength is given by the product of the production and decay signal strengths as µ(P, D) ≃ µ(P) µ(D) . ( Here P stands for one of the six productions processes of ggF, VBF, WH, ZH, ttH, and tH while D one of the six decay products of γγ, ZZ * , W W * , bb, τ τ , and µµ which are supposed to be visible at the LHC in this work.
The productions signal strength for the production process P is given by On the other hand, the decay signal strength for the process H → D is given by with the branching fraction of each decay mode defined by Note that an arbitrary non-SM contribution ∆Γ tot to the total decay width is introduced to parametrize invisible Higgs decays into non-SM and/or undetected particles. 11We observe that the partial and total decay widths of Γ(H → D) and Γ tot (H) becomes the SM ones of Γ(H SM → D) and Γ tot (H SM ), respectively, when g S H f f = 1, g P H f f = 0, g HW W,HZZ = 1, and ∆S γ,g = ∆P γ,g = 0. To calculate the decay widths of a generic Higgs boson, we meticulously follow the recent review on the decays of Higgs bosons [47] which provides explicit analytical expressions and supplemental materials for the individual partial decay widths as precisely as possible by including the state-of-the-art theoretical calculations of QCD corrections together with the SM electroweak corrections.
At leading order (LO), the six production signal strengths are given by µ(ggF in terms of the relevant form factors and couplings.For µ(VBF), we consider the W -and Z-boson fusion processes separately and the decomposition coefficients are given by the ratios of the SM cross sections of σ SM W W →H /σ SM VBF ≃ 0.73 and σ SM ZZ→H /σ SM VBF ≃ 0.27 which are largely independent of √ s.
For the tH production, we consider the two main production processes of q ′ b → tHq (tHq) and gb → tHW (tHW).In the 5-flavor scheme (5FS), the LO tHq process is mediated by the t-channel exchange of the W boson with H radiated from W or t.The LO tHW process is mediated by the s-channel exchange of the b quark with H, again, radiated from W or t.Both the production processes contain the two types of diagrams which involve the top-Yukawa and gauge-Higgs which gives µ(tH) in Eq. 19.
Note that the LO ggF production signal strength given in Eq. ( 19) should be reliable only when higher order corrections to the numerator and those to the denominator are largely canceled out in the ratio µ(ggF) = σ ggF /σ SM ggF .But, unfortunately, it turns out that the QCD corrections to the diagrams in which top quarks are running and those to the diagrams in which bottom quarks are running are significantly different [50] and, accordingly, the LO ggF production signal strength given in Eq. ( 19) is unreliable.In this work, combing ggF and bbH beyond LO, we use12 with the decomposition coefficients given in Table VIII.In comparison to Refs.[51,52], we exploit the N 3 LO (NNLO) production cross sections obtained by using SusHi-1.7.0 [53,54] for several combinations of the CP-even (CP-odd) Yukawa couplings having in mind the possibility that H is a CP-mixed state.We further consider the beyond-LO effects on the contributions from the triangle loops in which non-SM heavy particles are running.For the detailed derivation of the coefficients for µ(ggF) beyond LO in QCD, see Appendix A.  For the mixed-production mode involved with two production processes or more, we use the following production signal strength weighted by the cross sections of all the production processes involved: where, for the SM cross sections, we adopt those given in Ref. [55], see Table IX C.
Once all the theoretical signal strengths µ(Q, D) ≃ µ(Q) µ(D), each of which is associated with the specific production process of Q and the decay mode H → D, have been obtained, one may carry out a chi-square analysis.For the Tevatron data in which observables are uncorrelated, each χ 2 is given by where µ EXP (Q, D) and σ EXP (Q, D) denote the experimentally measured signal strength and the associated error, respectively.For the LHC Run 1 and Run 2 data, taking account of correlation among the observables in each set of data, we use χ 2 for n correlated observables: where the indices i, j count n correlated production-times-decay modes and ρ denotes the relevant n×n correlation matrix satisfying the relations of ρ ij = ρ ji and ρ ii = 1.If ρ ij = δ ij , we note that χ 2 n reduces to i.e., the sum of χ 2 of each uncorrelated observable.
For our chi-square analysis, we consider two statistical measures: (i) goodness of fit (gof) quantifying the agreement with the experimentally measured signal strengths in a given fit, and (ii) p-value against the SM for a given fit hypothesis to be compatible with the SM one: where n is the degree of freedom (dof) and m the number of fitting parameters against the SM null hypothesis with µ = 1.
In our case, we have n = 76 − m.The probability density function is given by with Γ(l/2) being the gamma function.The goodness of fit approaches to 1 when the value of χ 2 per degree of freedom becomes smaller.On the other hand, the p-value for compatibility with the SM hypothesis approaches to 1 when the test hypothesis becomes more and more SM-like.

IV. GLOBAL FITS
Now we are ready to perform global fits of the Higgs boson couplings to the full Higgs datasets collected at the LHC.Throughout this section, we use the following short notations for the 125 GeV Higgs H couplings to the SM particles: Depending on specific models, all the Higgs couplings are not independent.For example, the Higgs couplings to the massive vector bosons could be the same as in the SM and the Yukawa couplings could be the same separately in the upand down-quark and charged-lepton sectors.In this case, we denote the couplings as: Further, some of the Yukawa couplings could be the same like as in the four types of two Higgs doublet models (2HDMs) which are classified according to the Glashow-Weinberg condition [58] to avoid unwanted tree-level Higgs-mediated flavorchanging neutral currents (FCNCs).In this case, we denote the Higgs couplings as: 14 Last but not least, when all the Higgs couplings of C V and C S f to the SM particles scale with a single parameter as in, for example, Higgs-portal [60] and/or inert Higgs models, we denote the coupling as: 13 For the second statistical measure to test the SM null hypothesis with µ = 1, we use the likelihood ratio λ(1) = L(1)/L( µ): see Eq. (40.49) and below in the 2023 edition of the review "40.Statistics" by G. Cowan in Ref. [57].Note that, in the limit where the data sample is very large, the distribution of −2 ln λ(1) = χ 2 SM − χ 2 min approaches a χ 2 distribution with the number of degrees of freedom being equal to the number of fitting parameters. 14In this work, we adopt the conventions and notations of 2HDMs as in Ref. [59].
Our fits are categorized into the CP-conserving (CPC) and CP-violating (CPV) fits as in the previous studies [6,7,12].The CPC fits have been performed assuming that C P f = 0 and ∆P γ = ∆P g = 0 and, in this case, we have 10 varying fitting parameters which might be grouped into the non-SM and SM ones as follow: The non-SM parameters describe the variation of the signal strengths due to the H couplings to non-SM particles such as light invisible and heavy charged/colored ones.One the other hand, the SM parameters address the changes of the signal strengths due to the H couplings to the SM particles of the massive vector bosons (C V ) and the up-and down-type quarks (C S u,d ), and the charged leptons (C S ℓ ).When the normalized Yukawa couplings of H to the SM fermions are generation and flavor independent, we have only one Yukawa parameter To be more general, one may separately vary the H couplings to the W and Z bosons (C W and C Z ) without keeping the custodial symmetry between them and those to the top and charm quarks (C S t and C S c ).The couplings to a pair of tau leptons and muons (C S τ and C S µ ) also can be separately varied without assuming the lepton universality of the normalized charged-lepton Yukawa couplings.In the SM limit, the non-SM parameters are vanishing and all the SM ones take the SM value of 1.On the other hand, in the CPV fits under the assumption that H is a CP-mixed state, we have the following extended set of fitting parameters containing 17 parameters: A. CP-conserving fits We generically label the CPC fits as CPCn with n standing for the number of fitting parameters.Since there are 10 parameters to fit most generally, each CPCn contain several subfits.One can not exhaust all the possibilities and the CPC fits considered in this work are listed here: 15• CPC1: in this fit, we consider the four subfits as follows: -IU: vary ∆Γ tot to accommodate invisible Higgs decays into light non-SM and/or undetected particles -HC: vary ∆S γ to parametrize the contributions to H → γγ from the triangle loops in which heavy electrically charged non-SM particles are running -IH: vary C V f to address the case in which all the normalized Higgs couplings to the SM particles are the same like as in inert Higgs models -I: vary C S f to address the case in which all the normalized Yukawa couplings are the same like as type-I 2HDM • CPC2: in this fit, we consider the eight subfits as follows: -IUHC: vary {∆Γ tot , ∆S γ } for the case in which the light and heavy (electrically charged) non-SM particles coexist -HCC: vary {∆S γ , ∆S g } for the contributions to ggF, H → gg, and H → γγ from heavy non-SM particles which are electrically charged and colored -CSB: vary {C W , C Z } separately for the case in which the custodial symmetry between the W and Z bosons is broken -I: vary {C V , C S f } for the case in which all the Yukawa couplings are as in type-I 2HDM -II: vary {C S u , C S dℓ } for the case in which the Yukawa couplings are as in type-II 2HDM -III: vary {C S ud , C S ℓ } for the case in which the Yukawa couplings are as in type-III 2HDM -IV: vary {C S uℓ , C S d } for the case in which the Yukawa couplings are as in type-IV 2HDM -HP: vary {∆Γ tot , C V f } to address the Higgs-portal case in which Higgs decays invisibly and all the Higgs couplings to the SM particles scale with a single parameter • CPC3: in this fit, we consider the five subfits as follows: -IUHCC: vary {∆Γ tot , ∆S γ , ∆S g } for the case in which the light and heavy (charged and colored) non-SM particles coexist -II: vary {C V , C S u , C S dℓ } for the case in which the Yukawa couplings are as in type-II 2HDM -III: vary {C V , C S ud , C S ℓ } for the case in which the Yukawa couplings are as in type-III 2HDM -IV: vary {C V , C S uℓ , C S d } for the case in which the Yukawa couplings are as in type-IV 2HDM -HP: vary {∆Γ tot , ∆S γ , C V f } to address the Higgs-portal case when there exist heavy electrically charged non-SM particles in addition to light particles into which H could decay • CPC4: in this fit, we consider the two subfits as follows: -A: vary {C V , C S u , C S d , C S ℓ } for the case in which the Yukawa couplings are as in aligned 2HDM (A2HDM) [61] -HP: vary {∆Γ tot , ∆S γ , ∆S g , C V f } to address the Higgs-portal case when heavy charged and colored non-SM particles exist in addition to light particles into which H could decay • CPC5: in this fit, we consider the following two subfits:  f are varied as in CPC2-I.Also shown are the dashed and solid ellipses enclosing the 68% and 95% CL regions, respectively, presented in Refs.[13] (upper-left), [3] (upper-right), and [4] (lower-left).In the lower-right frame, the CL regions obtained using the full LHC Run 1 and Run 2 data are shown together with the solid magenta, blue, and black ellipses for the 95% CL regions presented in Refs.[13], [3], and [4] -LUB: vary {C V , C S u , C S d , C S τ , C S µ } to address the case in which the lepton universality of the normalized Yukawa couplings to charged leptons is broken • CPC6: in this fit, we consider the following scenario: to address the most general case of the Higgs couplings to the SM particles involved under the constraint of We provide Table X to summarize all the CPCn fits together with their subfits.Note that we do not address the case in which the charm-and top-quark Yukawa couplings are different from each other in this work.If only C S c is fitted while all the other non-SM and SM parameters are fixed at their SM values, we have |C S c | < ∼ 2 at 95% CL. 16 But, if other gauge-Higgs and Yukawa couplings are simultaneous varied taking C S c ̸ = C S t , fitting to the 76 signal strengths considered in this work does not lead to the bounded results for the couplings.From a search for the Higgs boson decaying into a pair of charm quarks, the ATLAS collaboration gives the observed (expected) constraints of |C S c | < 8.5 (12.4) at 95% CL and |C S c /C S b | < 4.5 at 95% CL (5.1 expected) [30].The CMS collaboration gives the observed (expected) 95% CL value of 1.1 < |C S c | < 5.5 (|C S c | < 3.40) [62].Before presenting the results of the CPCn fits and their subfits and discussing details of them separately, we make comparisons of the 68% and 95% confidence-level (CL) regions presented in Refs.[13], [3], and [4] with those obtained by 16 See Appendix B.
using the LHC Run 1 and Run 2 experimental signal strengths taken in this work 17 and the theoretical ones elaborated in subsection III B. To be specific, we have taken CPC2-I subfit in which the couplings C V and C S f are varied and the ATLAS⊕CMS Run 1, ATLAS Run 2, and CMS Run 2 CL regions are taken from Fig. 26 in Ref. [13], Fig. 4 in Ref. [3], and Fig. 3 in Ref. [4], respectively: see the regions inside the dashed (68%) and solid (95%) ellipses in the upper and lower-left frames of Fig. 1.The CL regions obtained in this work are colored in red (68%) and green (95%).We observe that the best-fit values for C S f agrees excellently for ATLAS Run 2 and CMS Run 2 while, for ATLAS⊕CMS Run 1, our value is smaller by the amount of about 0.04 which corresponds to about 0.5-σ level.On the other hand, our best-fit values for C V are nearer to the SM point and the differences are at the level below 0.7σ (Run 1) and 0.5σ (Run 2).The 1σ errors agree well except C V of ATLAS Run 2 for which we have obtained C this work V = 1.025 ± 0.025 while C ATLAS[3] V = 1.035 ± 0.031: the lower edges of the two 1σ regions are around 1 while our upper edge reduces to the SM direction by the amount 0.016 which corresponds to about 0.5-σ level.From these critical comparisons, we conclude that our global fits to the Higgs signal strengths in Tables II, III, and IV using the theoretical signal strengths given in subsection III B remarkably reproduces the fitting results in Ref. [13] (Run 1) and Refs.[3,4] (Run 2) within the 0.5-σ level.Further we have arrived at the conclusion that the combined results of our precision analysis of the full LHC Run 1 and Run 2 data should be reliable better than the 0.5-σ level since Run 2 data are now statistically dominant and our Run 2 results are more consistent with those in Refs.[3,4].Lastly, we present the fully combined results in the lower-right frame of Fig. 1: the 68% (red) and 95% (green) CL regions are obtained from the full LHC Run 1 and Run 2 data.For comparisons, we also show the 95% CL magenta, blue, and black solid ellipses from Refs.[13], [3], and [4], respectively, which are the same as in the upper-left, upper-right, and lower-left frames, respectively.From Run 1 to Run 2, we observe that C V approaches to the SM value of 1 while C S f deviates from it, see the points marked by pluses in the lower-right frame.The combined results gives C V = 1.015 ± 0.017 and C S f = 0.930 ± 0.031 and the SM point denoted by a star locates just outside of the 95% CL region.The deviation of C S f from its SM value of 1 has been noticed not only in Refs.[3,4] but also in Ref. [63] and our combined analysis strengthens the observation by showing that its best-fitted value is more than 2 standard deviations below the SM prediction.

CPC1
We show the fitting results for the four subfits of CPC1 and ∆χ 2 above each minimum in Table XI and Fig. 2, respectively.The p-values against the SM for compatibility with the SM hypothesis are high for IU and IH but they are only 8% (HC) and 1% (I).
In IU where we consider the case in which there exist light non-SM invisible and/or undetected particles and the 125   .Since the negative central value of ∆Γ tot is unphysical, we take the upper error of 0.287 MeV as the conservative upper limit to obtain the following limit on the non-SM branching ratio at 95% CL: which is better than the combined 95% CL limit of either 10.7% (ATLAS: 7.7% expected) [64] or 15% (CMS: 8% expected) [65] observed in searches for decays of the Higgs boson to invisible particles.
In HC where the gauge-Higgs and Yukawa couplings of the Higgs boson are the same as in the SM while there exist heavy electrically charged non-SM particles which could modify the loop-induced Higgs couplings to two photons, we obtain which shows a 1.8σ deviation from the SM.This could be understood by observing that the combined decay signal strength of the H → γγ mode is 1.10 ± 0.07, see Table VI.Note that S γ SM = −6.542+ 0.046 i and ∆S γ /S γ SM ≃ 0.048 ± 0.027.The gof value is 0.3474 which is definitely better than the SM.Incidentally, we obtain the following 95% CL region and limit: see the middle-left frame of Fig. 2.
In IH where all the Higgs couplings to the SM particles scale with a single coupling The gof value is the same as in CPC1-IU.Actually, in CPC1-IH and CPC1-IU, all the production and decay processes scale with the overall single theoretical signal strength as follows: Accordingly, the best-fitted values are consistent with the global signal strength of µ Global 76 signal strengths = 1.012 ± 0.034, see Eq. ( 6), which leads to the best-fit point deviated from the SM one by the amount of about +1% with about ±3% error in terms of signal strength.
In I where all the Yukawa couplings to the SM fermions scale with the same coupling parameter like as in type-I 2HDM but C V is fixed at its SM value of 1, we obtain Note that we have the highest gof value of 0.4753 which is larger than the CPC2-IV gof value of 0.4699 though slightly, see Table XII.We find that this simple one-parameter fit gives the best gof value among the CPC and CPV fits considered in this work, see Fig. 13.

CPC2
We show the fitting results for the eight subfits of CPC2 in Table XII and depict their CL regions in Fig. 3.In HP, we perform the fit under the constraints of ∆Γ tot ≥ 0 and C V f = C W = C Z ≤ 1 and, only for this, we have the gof value worse than the SM.Otherwise, the gof values range between 0.3267 (IUHC) and 0.4699 (IV) which are indeed better than the SM.The p-values against the SM for compatibility with the SM hypothesis are only a few % for I, II, III, and IV.
In IUHC, we assume the simultaneous existence of the light non-SM particles into which the Higgs boson H decays and the heavy electrically charged non-SM particles contributing to H → γγ through the triangle loops.The gof value is better than CPC1-IU but a little bit worse than CPC1-HC.Otherwise, the best-fitted values are similar to those in CPC1 with a bit larger 1σ errors.The SM point lies outside the 68% CL region, see the upper-left frame of the left panel of Fig. 3.
In HCC, we assume the existence of the heavy electrically charged and colored non-SM particles contributing to ggF and H → γγ through the triangle loops.The scalar form factor ∆S γ deviates from the SM by 2σ similarly as in CPC1-HC and CPC2-IUHC while ∆S g is consistent with the SM value of 0 within 1σ.The SM point lies outside of the 68% CL region, see the upper-right frame of the left panel of Fig. 3.The negative central value of −0.032 of ∆S g decreases |∆S g /S g SM | by the amount of about 5% with S g SM = 0.636 + 0.071 i which contributes to the 1% increment of the global signal strength µ Global 76 signal strengths since In CSB, C W is 2σ above the SM with the 1σ error of 2%, while C Z is consistent with the SM with the 1σ error of 3%.This is understood by comparing the WH production and H → W W * decay signal strengths of µ(WH, D) = 1.20±0.15and µ( P, W W * ) = 1.04 ± 0.07 to ZH production and H → ZZ * decay signal strengths of µ(ZH, D) = 1.03 ± 0.14   3.
In I, we assume all the Yukawa couplings to the SM particles are the same like as in type-I 2HDM but, compared to CPC1-I, C V is also varied.We obtain that While C V is consistent with the SM with the 1σ error of about 2%, C S f deviates from the SM by the amount of more than 2σ resulting in that the SM point lies just outside of the 95% CL region, see the lower-right frame of the left panel of Fig. 3.We observe that this is a combined result of µ( P, γγ) = 1.10 ± 0.07, µ( P, ZZ * ) = 0.97 ± 0.08, µ( P, W W * ) = 1.04 ± 0.07, µ( P, bb) = 0.90 ± 0.12, and µ( P, τ τ ) = 0.87 ± 0.08, see Table VI.More precisely, we find that the central value 1.10 of µ( P, γγ) correlates C V and C S f as C S f ∼ 3 C V − 2.1 19 under which the Yukawa coupling C S f is driven to give the branching ratios 10% below the SM by µ( P, bb) and µ( P, τ τ ) while the gauge coupling C V near to the SM value of 1 by µ( P, ZZ * ) and µ( P, W W * ).Indeed, we find that by fitting to the γγ signal strengths only and by fitting to the fermionic signal strengths only.Incidentally, we obtain by fitting to the bosonic signal strengths only.
In II, III, and IV, taking C V = 1, we assume the Yukawa couplings to the SM particles behave like as in type-II, We note that C S ℓ is basically determined by µ( P, τ τ ) = 0.87 ± 0.08, see Table VI.Otherwise, all the Yukawa couplings deviate from the SM by the amount of more than 2σ like as in CPC2-I with the 1σ errors of 3%-4%.The SM point lies around the boundary between the 95% and 99.73% CL regions, see the upper-left (II), upper-right (III), and lower-left (IV) frames of the right panel of Fig. 3.We further note that all the best-fitted values are positive and the negative values of the Yukawa couplings C S dℓ , C S ℓ , and C S d around −1 are a bit less favored.In Fig. 4, we show ∆χ 2 above the positive minimum versus the down-type Yukawa couplings in the II (left), III, (middle), and IV (right) subfits.For C S dℓ , C S ℓ , and C S d , we observe that the data prefer the positive minima to the negative ones by ∆χ 2 ∼ 1.5 (C S dℓ and C S d ) and ∆χ 2 ∼ 0.5 (C S ℓ ), see the upper frames of Fig. 4.This could be understood by observing that µ(ggF + bbH) increases by the amount of about 10% by changing C S b from +1 to −1, see Eq. ( 22) and Table VIII. 20Similarly, µ(γγ) is also sensitive to the sign of C S τ but the sign dependence is weaker due to the dominance of the W -boson loop contribution to S γ , see Eq. (11).In fact, µ(γγ) is powerful to reject the wrong sign of the top-quark Yukawa coupling and we see that negative C S t is completely ruled out, see the lower frames of Fig. 4. In II, III, and IV, we scrutinize that the fitting results are consistent with the pattern of the Yukawa couplings predicted in each model.In type-II, type-III, and type-IV 2HDMs, the Yukawa couplings are correlated as follows: 21

II :
C S u = cos γ − 1/ tan β sin γ , C S dℓ = cos γ + tan β sin γ , III : Note that, in each model, only one of the two couplings could be larger or smaller than 1 depending on the sign of sin γ when cos γ = C V ∼ 1.It is impossible to have both the couplings larger or smaller than 1 likes as in type-I 2HDM.In the upper-left (II), upper-right (III), and lower-left (IV) frames of the right panel of Fig. 3, we note that most of the 95% CL regions locate where both of the couplings are smaller than 1.The situation will be clearer in CPC3 by varying C V also and in CPC4-A by varying the Yukawa couplings of the up-and down-type quarks and the charged leptons separately.
In HP, all the production and decay processes scale with the overall single theoretical signal strength of: which leads to the relation In the lower-right frame of the right panel of Fig. 3, the black line passing the origin (∆Γ tot , C V f ) = (0, 0) and the SM point (∆Γ tot , C V f ) = (0, 1) represents the above relation when µ Global = 1.In Higgs-portal models, the varying parameters are physically constrained by ∆Γ tot ≥ 0 and C V f ≤ 1. Imposing these conditions, we find the following best-fitted values We consider some extended HP scenarios in CPC3 and CPC4.

CPC3 and CPC4
We show the fitting results for the five CPC3 and two CPC4 subfits in Table XIII.In the HP scenarios, we perform the fit under the constraints of ∆Γ tot ≥ 0 and C V f = C W = C Z ≤ 1 and we have the gof values similar to the SM.Otherwise, the gof values range between 0.3181 (CPC3-IUHCC) and 0.4377 (CPC3-II) which are better than the SM but slightly worse than the corresponding CPC2 fits.The gof value of CPC4-A is also larger than 0.4.The CL regions in two-parameter planes are depicted in Fig. 5 and Fig. 6 for CPC3 and CPC4, respectively.Note that, in the HP scenarios, we show the parameter spaces in which the fitting constraints of ∆Γ tot ≥ 0 and C V f ≤ 1 are fulfilled.The pvalues against the SM for compatibility with the SM hypothesis are smaller than 10% except CPC3-IUHCC, CPC3-HP, and CPC4-HP.
In CPC3-IUHCC, we vary all the three non-SM parameters.The best-fitted values for ∆Γ tot and ∆S g are consistent with the SM within 1σ while ∆S γ deviates from the SM by about 1.5σ.The 1σ errors are about 5% for ∆Γ tot /Γ tot (H SM ), 3% for |∆S γ /S γ SM |, and 6% for |∆S g /S g SM | which are slightly larger than those found in CPC1-IU, CPC2-IUHC, and CPC2-HCC.The SM points lies outside of the 68% CL regions in the (∆S γ , ∆Γ tot ) and (∆S g , ∆S γ ) planes, see the upper frames of the left panel of Fig. 5.
In II, III, IV subfits of CPC3, we additionally vary C V compared to the corresponding CPC2 subfits.We observe that C V is consistent with SM within 1σ errors of about 2%-3%.In contrast, the Yukawa couplings are about 2σ below the SM except for C  (C S u , C V ) (II) plane appearing the upper-left frame of the right panel of Fig. 5 is due to the minimum around C S u = 0.9 for the negative values of C S dℓ , see the upper-right frame of the same panel in the (C S dℓ , C S u ) plane.Finally, we observe that the 68% CL regions locate where both of the Yukawa couplings are smaller than 1 indicating deviation from the conventional type-II, type-III, and type-IV 2HDMs, see the upper-, middle-, and lower-right frames of the right panel of Fig. 5.
In CPC3-HP where we add the non-SM contribution to H → γγ compared to CPC2-HP, ∆S γ is fitted to accommodate µ(γγ) = 1.1±0.07like as in CPC1-HC.The parameters ∆Γ tot and C V f are fitted to have the SM values like as in CPC2-HP but with a bit larger 1σ errors under the constraints of ∆Γ tot > 0 and C V f < 1.In CPC4-HP, we further add the non-SM contribution also to H → gg assuming non-SM particles such as vector-like quarks.For ∆S γ and ∆S g , the fitting results are very similar to CPC2-HCC and the parameters ∆Γ tot and C V f are again fitted to have the SM values under the constraints of ∆Γ tot > 0 and C V f < 1 like as in other HP scenarios.The SM points lie outside of the 68% CL regions in the (∆Γ tot , ∆S γ ), (C V f , ∆S γ ), and (∆S g , ∆S γ ) planes, see the right panel of Fig. 6.
In CPC4-A, we vary the Yukawa couplings of the up-and down-type quarks and the charged leptons separately together with C V .This scenario does not alter our previous observation made in CPC3 for the Yukawa couplings: they are about 1σ (C S d ) and 2σ (C S u and C S ℓ ) below the SM.The C V is very consistent with the SM with the 1σ error of about 3% and the 1σ errors of C S u , C S ℓ , and C S d are 4%, 5%, and 8%, respectively.And, from the CL regions in the (C S d , C S u ), (C S ℓ , C S u ), and (C S ℓ , C S d ) planes shown in the lower frames of the left panel of Fig. 6, we see that the data favor the type-I 2HDM over the other three models.Incidentally, we find that the minima for the negative values of C S d and C S ℓ are above the positive ones by the amount of ∆χ 2 ∼ 1.5 and ∼ 0.3, respectively.

CPC5 and CPC6
We show the fitting results for the two CPC5 and one CPC6 subfits in Table XIV.The gof values are 0.3809 (CPC5-AHC), 0.3972 (CPC5-LUB) and 0.3803 (CPC6-CSBLUB) which are better than the SM.The p-values against the SM for compatibility with the SM hypothesis are low and it is 15% for CPC6-CSBLUB.The CL regions in two-parameter planes are depicted in Fig. 7 and Fig. 8 for CPC5 and CPC6, respectively.
In CPC5-AHC, compared to CPC4-A, we add the contribution to H → γγ from heavy electrically charged particles.First of all, we find that the minima for the positive and negative values of C S ℓ are degenerate with the change of ∆S γ by the amount of 0.044 compensating the effects of the flipped sign of C S ℓ , see Eq. (11).The parameter ∆S γ is consistent with the SM: ∆S γ /S γ SM ∼ 0.016 ± 0.03 and 0.022 ± 0.03 for the positive and negative values of C S ℓ , respectively, and the two minima of ∆S γ are very near to each other separated by only ∼ 0.2σ.The gauge-Higgs coupling C V and the Yukawa couplings of C S u , C S d , and |C S ℓ | are fitted similarly as in CPC4-A. 22The SM points are now near to or in the 68% CL regions, see the left panel of Fig. 7.We find that the negative minimum of C S d is above the positive one by the amount of ∆χ 2 ∼ 1.5.

FIG. 8. CPC6:
The CL regions of CPC6-CSBLUB.The contour regions shown are for ∆χ 2 ≤ 2.3 (red), ∆χ 2 ≤ 5.99 (green), ∆χ 2 ≤ 11.83 (blue) above the minimum, which correspond to confidence levels of 68.27%, 95%, and 99.73%, respectively.In each frame, the vertical and horizontal lines locate the SM point denoted by a star and the best-fit point is denoted by a triangle.
In CPC5-LUB, compared to CPC4-A, we fit the tau-lepton-and muon-Yukawa couplings separately.We obtain that the gauge-Higgs coupling is consistent with the SM with the 1σ error of about 3% and the up-quark and tau-lepton Yukawa couplings are about 2σ below the SM with the 1σ errors of about 4%.The down-quark Yukawa coupling is about 1σ below the SM with the 1σ error of about 7%-8%.We find that the minima for the positive and negative values of C S µ are degenerate and, at the positive minimum, the muon-Yukawa coupling is consistent with the SM with the 1σ error of 13%-15%.The best-fitted values of the gauge-Higgs coupling C V and the Yukawa couplings of C S u,d,τ are the same at the two generate minima.Around the SM values of C S τ = C S µ = 1, the 1σ regions of C S τ 1σ = [0.866, 0.952] and C S µ 1σ = [0.906, 1.191] overlap with no violation of lepton universality.We find that the negative minima of C S d and C S τ TABLE XV.Varying parameters in the CPVn fits and their subfits considered in this work.The parameters not mentioned are supposed to take the SM value of either 0 or 1.For the total 17 CPV parameters, see Eq. (32).
are above the positive ones by the amount of ∆χ 2 ∼ 1.5 and ∼ 0.2, respectively.The negative and positive regions of C S µ are connected at 99.73% CL, see the right panel of Fig. 7.
In CPC6-CSBLUB, we vary the six SM parameters independently under the constraint of C S t = C S c = C S u and we find that the gauge-Higgs couplings are consistent with the SM with the 1σ errors of about 3%-4%.The central value of C W (C Z ) is slight above (below) the SM value of 1.The up-quark and tau-lepton Yukawa couplings are about 2σ below the SM with the 1σ errors of about 4%.The down-quark Yukawa coupling is about 1σ below the SM with the 1σ error of about 7%.We find that the minima for the positive and negative values of C S µ are degenerate and, at the positive minimum, the muon Yukawa coupling is consistent with the SM with the 1σ error of about 13%.Before moving to CPV, we provide the following brief summary for the SM parameters obtained from CPC3, CPC4, CPC5, and CPC6 fits: 23• C V , C W , C Z : consistent with the SM with the 1σ error of 2%-3% • C S u , C S ud , C S uℓ : about 2σ below the SM with the 1σ error of 3%-4% • C S dℓ , C S ℓ , C S τ : about 2σ below the SM with the 1σ error of 4%-5% • C S d : about 1σ below the SM with the 1σ error of 7%-8% • |C S µ |: consistent with the SM with the 1σ error of 12-15% We further note that the BSM models predicting the same normalized Yukawa couplings to the up-and down-type quarks and charged leptons are preferred.

B. CP-violating fits
We generically label the CPV fits as CPVn with n standing for the number of fitting parameters like as in the CPC fits.Since there are 17 parameters to fit most generally, it is more challenging to exhaust all the possibilities than in the CPC fits.Noting that CP violation is signaled by the simultaneous existence of the Higgs couplings to the scalar and pseudoscalar fermion bilinears, 24 we consider the following CPV fits in this work: • CPV2: in this fit, we consider the four subfits as follows: -U: vary {C S u , C P u } for the case in which CP violation resides in the up-type quark sector -D: vary {C S d , C P d } for the case in which CP violation resides in the down-type quark sector -L: vary {C S ℓ , C P ℓ } for the case in which CP violation resides in the charged-lepton sector -HC: vary {∆S γ , ∆P γ } for the case in which CP violation occurs due to heavy electrically charged non-SM fermions coupling to the Higgs boson • CPV3: in this fit, we consider the five subfits as follows: -U: vary {C V , C S u , C P u } for the up-quark sector CP violation -D: vary {C V , C S d , C P d } for the down-quark sector CP violation -L: vary {C V , C S ℓ , C P ℓ } for the charged-lepton sector CP violation -F: vary {C V , C S f , C P f } assuming the universal normalized CPV couplings to the SM quarks and charged leptons -IUHC: vary {∆Γ tot , ∆S γ , ∆P γ } for the case in which CP violation occurs due to the H couplings to heavy electrically charged non-SM fermions in the presence of light non-SM particles into which H could decay • CPV4: in this fit, we consider the following two subfits: -IUF: vary {∆Γ tot , C V , C S f , C P f } assuming the universal normalized CPV couplings to the SM fermions in the presence of light non-SM particles into which H could decay -HCC: vary {∆S γ , ∆P γ , ∆S g , ∆P g } for the case in which CP violation occurs due to the H couplings to heavy electrically charged and colored non-SM fermions • CPV5: in this fit, we consider the following scenario: -IUHCC: vary {∆Γ tot , ∆S γ , ∆P γ , ∆S g , ∆P g } for the case in which CP violation occurs due to the H couplings to heavy electrically charged and colored non-SM fermions in the presence of light non-SM particles into which H could decay • CPV7: in this fit, we consider the following scenario: with the H couplings to the SM particles like as in CPV A2HDM We provide Table XV for the summary of the CPV fits considered in this work which explicitly shows the parameters varied in each subfit of CPVn.
Since the signal strengths are CP-even quantities, they do not contain CPV products such as C S u,d,ℓ,f × C P u,d,ℓ,f and S γ,g ×P γ,g .Therefore, the CL regions appear as a circle or an ellipse or some overlapping of them in the (C S u,d,ℓ,f , C P u,d,ℓ,f ) and (∆S γ,g , ∆P γ,g ) planes.

CPV2 and CPV3
We show the fitting results for the four CPV2 and five CPV3 subfits in Table XVI.We have the largest gof value for CPV3-F and note that the p-values against the SM for compatibility with the SM hypothesis are high in CPV2-U and CPV2-D with χ 2 min ∼ χ 2 SM .In the left panel of Fig. 9, the CL regions are depicted in the (C S u,d,ℓ,f , C P u,d,ℓ,f ) and (∆S γ , ∆P γ ) planes for CPV2 and CPV3.The other CPV3 CL regions in the (C S,P u,d,ℓ,f , C V ), (∆S γ , ∆Γ tot ), and (∆P γ , ∆Γ tot ) planes are shown in the right panel of Fig. 9.
In CPV2-U, we obtain the sickle-shaped CL region in the (C S u , C P u ) plane, see the upper-left frame of the left panel of Fig. 9.This could be understood by observing that the top-Yukawa couplings are involved in the ggF+bbH and ttH⊕tH production processes and the H → γγ decay mode.From Eq. ( 22) with the Run 2 decomposition coefficients in Table VIII, Eq. ( 11) with S γ SM = −6.542+ 0.046 i, and Eq. ( 24), we have where the factor 1.016 in the second line for µ(γγ) takes account of the difference in the QCD and electroweak corrections to the scalar and pseudoscalar parts [47].Note that µ(ggF + bbH) ≃ 1 gives an ellipse centered at (C S u , C P u ) ≃ (0, 0) with the lengths of the major (C S u ) and minor (C P u ) axes of 1 and 0.67 while µ(γγ) ≃ 1 an ellipse centered at (C S u , C P u ) ≃ (4.6, 0) with the lengths of the major (C S u ) and minor (C P u ) axes of 3.6 and 2.4.In addition, µ(ttH⊕tH) ≃ 1 gives a circle centered at (C S u , C P u ) ≃ (0.3, 0) with a radius of about 0.7.Both the ellipses and the circle pass the SM point of (C S u , C P u ) = (1, 0) as they should with the ggF+bbH ellipse and the ttH⊕tH circle extending to the negative C S u direction from the SM point and the γγ ellipse to the positive C S u direction.The overlapping of the two ellipses and a circle with some corresponding   errors explain the sickle-shaped CL region in the (C S u , C P u ) plane which also appears in CPV3-U and CPV3-F, see the middle-left and lower-left frames of the left panel of Fig. 9.We observe that the SM point lies outside the 68% CL region in CPV3-F with C S f = 0.930 +0.031 −0.081 which deviates from the SM point more than C S u in CPC2-U and CPC3-U with the smaller negative error.
The circles in the (C S d , C P d ) planes for CPV2-D and CPV3-D shown in the two middle-left frames of the left panel of Fig. 9 are understood by noting that the signal strength of the bb decay mode is given by µ The positive values of C S d are preferred because of the interferences between the top-and bottom-quark contributions to ggF.We note that ∆χ 2 above the minimum at (C S d , C P d ) ≃ (1, 0) increases by the amount of about 5 while C S d changes from +1 to −1.When C P d changes from +1 to −1, ∆χ 2 above the minimum increases by the amount smaller than 2. The circles in the (C S ℓ , C P ℓ ) planes for CPV2-L and CPV3-L shown in the two middle-right frames of the left panel of Fig. 9 are understood by noting that the signal strength of the τ τ and µµ decay modes is given by µ(τ τ ) ≃ µ(µµ) ≃ (C S ℓ ) 2 + (C P ℓ ) 2 .We note that ∆χ 2 above the minimum (C S ℓ , C P ℓ ) ≃ (0.94, 0) increases by the amount less than 1 while C S,P ℓ changes from +1 to −1.We note that the charged-lepton circles are smaller than the down-type-quark circles.
The circles in the (∆S γ , ∆P γ ) planes for CPV2-HC and CPV3-IUHC shown in the two right frames of the left panel of Fig. 9 are understood by noting that µ(γγ) = 1 gives a circle centered at (∆S γ , ∆P γ ) ≃ (6.5, 0) with the radius of about 6.5 with the signal strength of the H → γγ decay mode given by From the ten frames of the right panel of Fig. 9, we observe that the most of the SM points are outside of the 68% CL regions except CPV3-F in the (C P f , C V ) plane (middle-right) and CPV3-IUHC (lower).There are almost no correlations between C V and C S,P ℓ (upper-middle-right and middle-middle-right) and the correlation between C V and C P d in CPV3-D (middle-middle-left) is weakly correlated.We also see almost no correlations between ∆Γ tot and ∆S γ (lower-left) and ∆Γ tot and ∆P γ (lower-middle) in CPV3-IUHC.

CPV4, CPV5, and CPV7
We show the fitting results for CPV4, CPV5, and CPV7 in Table XVII.Note that, in CPV4-IUF, the parameters are not bounded like as in the HP scenarios and we implement the fit under the constraints of ∆Γ tot ≥ 0 and C V ≤ 1.We have the largest gof value for CPV4-IUF which is slightly higher than that of CPV3-F.The left panel of Fig. 10 is for CPV4.In the six left and middle frames, the CL regions in IUF are depicted and, in the two upper-and middle-right frames, those in HCC are shown in the (∆S γ , ∆P γ ) and (∆S g , ∆P g ) planes.The right panel of Fig. 10 is for CPV5-IUHCC and the CL regions in the (∆S γ , ∆P γ ) and (∆S g , ∆P g ) planes are depicted in the two left frames.In Fig. 11, we show the CL regions of CPV7 in the (C S u,d,ℓ , C P u,d,ℓ ) planes in the upper three frames and some others below them.In CPV4-IUF, ∆Γ tot and C V are driven to the SM values under the constraints of ∆Γ tot ≥ 0 and C V ≤ 1 and, being different from CPV2 and CPV3, we have the two degenerate minima at (C S f , C P f ) ≃ (0.90, ±0.17).The central value of C S f is smaller than those in the CPC fits by the amount of 2%-3% which is compensated by the relation of (C S f ) 2 + (C P f ) 2 ≃ 0.92 at the minima.The CL regions are shown in the left and middle six frames of the left panel of Fig. 10.Both the two degenerate minima are in the 68% CL region and the SM point is outside the 95% CL region, see the left three frames for C P f , ∆Γ tot , and C V versus C S f .
In CPV4-HCC, we have the µ(γγ) circle centered at (∆S γ , ∆P γ ) ≃ (7, 0) with the radius of about 7, see the upperright frame of the left panel of Fig. 10.There is no visible change in ∆χ 2 above the minimum along the circle passing the center of the 68% CL region.On the other hand, we obtain the sickle-shaped CL region in the (∆S g , ∆P g ) plane, see the middle-right frame of the left panel of Fig. 10.This is understood by the overlapping of the µ(ggF + bbH) and µ(D ̸ = γγ, gg) circles: For µ(ggF+bbH), we use Eq. ( 22) with the Run 2 decomposition coefficients in Table VIII.We note that µ(ggF+bbH) = 1 gives a circle centered at (∆S g , ∆P g ) ≃ (−1.15, 0) with the radius of about 1.15.In the second line for µ(D ̸ = γγ, gg), the factor 0.96 takes account of the difference in the QCD and electroweak corrections to the scalar and pseudoscalar parts [47].With S g SM = 0.636 + 0.071 i and P g SM = 0, we note that µ(D ̸ = γγ, gg) = 1 gives a circle centered at (∆S g , ∆P g ) ≃ (−0.64, 0) with the radius of about 0.64 which is smaller than the µ(ggF + bbH) circle.Note that we obtain the sickle-shaped CL region in the (∆S g , ∆P g ) plane because we consider the ggF+bbH production process beyond LO in QCD.[Right] The CL regions of CPV5-IUHCC in two-parameter planes.The contour regions shown are for ∆χ 2 ≤ 2.3 (red), ∆χ 2 ≤ 5.99 (green), ∆χ 2 ≤ 11.83 (blue) above the minimum, which correspond to confidence levels of 68.27%, 95%, and 99.73%, respectively.In each frame, the vertical and horizontal lines locate the SM point denoted by a star and the best-fit point is denoted by a triangle.

FIG. 11. CPV7:
The CL regions of CPV7-A in two-parameter planes.The contour regions shown are for ∆χ 2 ≤ 2.3 (red), ∆χ 2 ≤ 5.99 (green), ∆χ 2 ≤ 11.83 (blue) above the minimum, which correspond to confidence levels of 68.27%, 95%, and 99.73%, respectively.In each frame, the vertical and horizontal lines locate the SM point denoted by a star and the best-fit point is denoted by a triangle.
In CPV5-IUHCC, compared to CPV4-HCC, we vary ∆Γ tot additionally.The best-fitted values are similar to those in CPV4-HCC with a bit larger errors for ∆S g and ∆P g .We observe that the sickle-shaped CL region in the (∆S g , ∆P g ) plane extends at the cost of negative ∆Γ tot , see the middle three frames in the right panel of Fig. 10.Note that ∆Γ tot is almost insensitive to ∆S γ and ∆P γ in the allowed regions.-A).For the pseudoscalar couplings, C P u is constrained around its SM value of 0 with the 1σ error of 30%.For the other pseudoscalar couplings of C P d and and C P ℓ , we have the 1σ errors of about 100%.We clearly see the SM points outside of the 68% CL in the (C S u , C P u ) and (C S u , C V ) planes, see the upper-and middle-left frames of Fig. 11.

C. Predictions for H → Zγ
Recently, the ATLAS and CMS collaborations report the first evidence for the Higgs boson decay to a Z boson and a photon with a statistical significance of 3.4 standard deviations based on the Run 2 data with 140/fb luminosity for each experiment [14].The combined analysis gives the measured signal yield of 2.2 ± 0.7 times the SM prediction which corresponds to B(H → Zγ) = (3.4± 1.1) × 10 −3 assuming SM Higgs boson production cross sections. 25he loop-induced Higgs couplings to a Z boson and a photon are similarly described as those to two photons and two gluons by using the scalar and pseudoscalar form factors of S Zγ and P Zγ .For the detailed description and analytic structure of them, we refer to Ref. [47].Taking M H = 125 GeV, we have retaining only the dominant contributions from third-generation SM fermions and the charged gauge bosons W ± and introducing ∆S Zγ and ∆P Zγ to parametrize contributions from the triangle loops in which non-SM charged particles are running.In the SM limit, S Zγ SM = −11.6701+ 0.0114 i and P Zγ SM = 0.
We first examine how large µ(Zγ) can be in CPC2-I in which C V and C S f are varied in the absence of non-SM particles contributing to ∆S Zγ .In this scenario, we have where we use Eq.(C. leading to the enhanced Higgs decay into Zγ by the amount of 30% at the upper boundary of the 95% CL region which is in the right direction to be consistent with the the measured signal strength of 2.2 ± 0.7.We note that C S f fitted below the SM value of 1 increases µ(Zγ), see the two CPC2-I frames of Fig. 12.In CPC4-A and CPC6-CSBLUB where we have the more fitting parameters of the gauge-Higgs and Yukawa couplings but still with ∆S Zγ = 0, we find that µ(Zγ) CPC4−A ≃ µ(Zγ) CPC6−CSBLUB ≃ 1.17

V. CONCLUSIONS
We perform global fits of the Higgs boson couplings to the full Higgs datasets collected at the LHC with the integrated luminosities per experiment of approximately 5/fb at 7 TeV, 20/fb at 8 TeV, and up to 139/fb at 13 TeV.To enhance the sensitivity of our global analysis, we combine the LHC Run 1 dataset with the two Run 2 datasets separately given by the ATLAS and CMS collaborations ignoring correlations among them.We have carefully chosen the 76 production-timesdecay signal strengths and, based on them, we consistently reproduce the global and individual (production and decay) signal strengths in the literature.We further demonstrate that our combined analysis based on the 76 experimental signal strengths and the theoretical ones elaborated in this work reliably reproduce the fitting results presented in Ref. [13] (Run 1) and Refs.[3,4] (Run 2) within 0.5 standard deviations.Note that we have included the production signal strength for the tH process to accommodate the new feature of the LHC Run 2 data and considered the ggF production process beyond leading order in QCD to match the level of precision of the LHC Run 2 data.
We have implemented the 22 CPC subfits from CPC1 to CPC6 in Table X and the 13 CPV subfits from CPV2 to CPV7 in Table XV taking account of various scenarios found in several well-motivated BSM models.Our extensive and comprehensive analysis reveals that the LHC Higgs precision data are no longer best described by the SM Higgs boson. 26For example, in CPC2-I for which we obtain the higher gof value of 0.47 than in the SM and the low p-value of 0.02 for compatibility with the SM, we find the following best-fitted values of with C V being consistent with the SM with the 1σ error of 2% and C S f below the SM by more than 2 standard deviations with the 1σ error of 3%.We show that this could be understood by looking into the individual decay signal strengths presented in Table VI 1 around the SM point under which W W * and ZZ * drive C V near to the SM value of 1 while the Yukawa couplings are driven smaller to match the signal strengths of about 0.9 for pp → H → bb and pp → H → τ τ .In CPC3, CPC4, CPC5, and CPC6 where we have the more fitting parameters of the gauge-Higgs and Yukawa couplings, we find that these features remain the same but with a bit larger 1σ errors.Explicitly, we observe the following behavior of the gauge-Higgs and Yukawa couplings to the SM particles: • C V , C W , C Z : consistent with the SM with the 1σ error of 2%-3% • C S u , C S ud , C S uℓ : about 2σ below the SM with the 1σ error of 3%-4% • C S dℓ , C S ℓ , C S τ : about 2σ below the SM with the 1σ error of 4%-5% • C S d : about 1σ below the SM with the 1σ error of 7%-8% • C S µ : consistent with the SM with the 1σ error of 12-15% Incidentally, in many of the two-parameter planes, the SM points locate outside the 68% CL region easily and even the 95% CL region sometimes.In Fig. 13, we compare the gof values of all the CPCn and CPVn subfits considered in this work.We indeed observe that the most of them have the better goodness of fit than the SM.Incidentally, we note that CP violation is largely unconstrained by the LHC Higgs data with the CL regions appearing as a circle or an ellipse or some overlapping of them in the CP-violating two-parameter planes.We explain the details of how the ellipses and circles emerge in several subfits of CPVn.Especially, in CPV4-HCC and CPV5-IUHCC, we note that the sickle-shaped CL regions in the (∆S g , ∆P g ) plane are obtained since we consider the ggF production beyond LO in QCD, Interestingly, we find that the BSM models predicting the same normalized Yukawa couplings to the up-and downtype quarks and charged leptons are preferred.For example, among the four types of 2HDMs classified according to the Glashow-Weinberg condition to avoid FCNCs, this could be achieved only in the type-I 2HDM.Last but not least, we note that the reduced Yukawa couplings help to explain the combined H → Zγ signal strength of 2.2 ± 0.7 recently reported by the ATLAS and CMS collaborations [14].But one might need nonvanishing ∆S Zγ ∼ −5 to comfortably accommodate the large central value of 2.2.assuming that ∆S g and ∆P g are real and the interferences terms proportional to the products of g S,P H bb × g S,P H cc , g S H bb ,H cc × ∆S g , g P H bb ,H cc × ∆P g and the diagonal terms (g S,P H cc ) 2 have been neglected.
To go beyond LO in QCD, to begin with, we consider the contributions from top-, bottom-, and charm-quark loops taking ∆S g = ∆P g = 0.In this case, the ggF production cross section of a CP-mixed Higgs boson H might be organized as follow:  assuming that ∆P g is real, taking P g tt = g P H tt in the infinite M t limit, and using ℜe 1 + ϵ P tt ≤ |1 + ϵ P tt | = σ P tt / σ P tt LO .
In Table XIX, we show the cross sections σ S,P t∆ and σ S,P ∆∆ in pb at √ s = 7 TeV, 8 TeV, and 13 TeV assuming ∆S g and ∆P g are real.We use the values of the cross sections σ S,P tt = σ S,P ggF (1, 0, 0) and (σ S,P tt ) LO in Table XVIII together with the relations given by Eqs.(A.16 and (A.19).For σ S,P t∆ , we take the approximation ℜe(1 + ϵ S,P tt ) ≈ 1 + ϵ S,P tt and, for the contributions from the triangle top-quark loops, we take the M t → ∞ limit.Then, with the cross sections σ S,P t∆ and σ We again note that the decomposition coefficients are almost independent of √ s and, comparing with the LO result Eq. (A.3), we observe that the coefficients proportional to the products of (g S H tt )(∆S g ) and (g P H tt )(∆P g ) and the squares of (∆S g ) 2 and (∆P g ) 2 decrease by factors of about 2 to 3.   where we use µ( P, γγ) = 1.10 ± 0.07, see Table VI.Note that the above relation is equivalent to C S f ∼ 3 C V − 2.1 for the central value of 1.1 which is quoted below Eq. ( 41).Since one might need nonvanishing ∆S Zγ to resolve the tension in the measured H → Zγ signal strength as discussed in subsection IV C, we consider HC scenarios in which there exist heavy electrically charged non-SM particles leading to nonvanishing ∆S γ and ∆S Zγ simultaneously.To be specific, we consider the following two CPC fits: • CPC2-HC: vary {∆S γ , ∆S Zγ } with the gauge-Higgs and Yukawa couplings the same as in the SM • CPC6-AHC: vary {∆S γ , ∆S Zγ , C V , C S u , C S d , C S ℓ } with the gauge-Higgs and Yukawa couplings like as in A2HDM Note that we have promoted CPC1-HC and CPC5-AHC by employing ∆S Zγ as an additional varying parameter.
In CPC2-HC, we obtain the following best-fitted values and 1σ errors:  with χ 2 min /dof = 79.1649/75,gof = 0.3489, and p-value against the SM = 0.0468.We observe that ∆S γ is fitted similarly as in CPC1-HC, see Table XI, and there are two degenerate minima for the negative (< 0) and positive (> 0) values of type -IV 2HDM) .
like as in CPC4-A in the presence of heavy electrically charged particles such as charged Higgs bosons contributing to H → γγ

FIG. 1 .
FIG. 1.The 68% (red) and 95% (green) CL regions in the (CV , C S f ) planes obtained using the ATLAS⊕CMS Run 1 (upper-left), ATLAS Run 2 (upper-right), and CMS Run 2 (lower-left) experimental signal strengths in Tables II, III, and IV, respectively, and the theoretical ones in subsection III B. The couplings of CV and C Sf are varied as in CPC2-I.Also shown are the dashed and solid ellipses enclosing the 68% and 95% CL regions, respectively, presented in Refs.[13] (upper-left),[3] (upper-right), and[4] (lower-left).In the lower-right frame, the CL regions obtained using the full LHC Run 1 and Run 2 data are shown together with the solid magenta, blue, and black ellipses for the 95% CL regions presented in Refs.[13],[3], and[4].The colors and lines are the same in all the frames and the vertical and horizontal lines denote the SM values of CV = 1 and C S f = 1 with the best-fit points denoted by triangles (colored regions) and pluses (ellipses).The SM points where CV = C S f = 1 are denoted by stars.
FIG. 1.The 68% (red) and 95% (green) CL regions in the (CV , C S f ) planes obtained using the ATLAS⊕CMS Run 1 (upper-left), ATLAS Run 2 (upper-right), and CMS Run 2 (lower-left) experimental signal strengths in Tables II, III, and IV, respectively, and the theoretical ones in subsection III B. The couplings of CV and C Sf are varied as in CPC2-I.Also shown are the dashed and solid ellipses enclosing the 68% and 95% CL regions, respectively, presented in Refs.[13] (upper-left),[3] (upper-right), and[4] (lower-left).In the lower-right frame, the CL regions obtained using the full LHC Run 1 and Run 2 data are shown together with the solid magenta, blue, and black ellipses for the 95% CL regions presented in Refs.[13],[3], and[4].The colors and lines are the same in all the frames and the vertical and horizontal lines denote the SM values of CV = 1 and C S f = 1 with the best-fit points denoted by triangles (colored regions) and pluses (ellipses).The SM points where CV = C S f = 1 are denoted by stars.

FIG. 3 .
FIG. 3. CPC2:The CL regions of the eight CPC2 subfits in two-parameter planes.The contour regions shown are for ∆χ 2 ≤ 2.3 (red), ∆χ 2 ≤ 5.99 (green), ∆χ 2 ≤ 11.83 (blue) above the minimum, which correspond to confidence levels of 68.27%, 95%, and 99.73%, respectively.In each frame, the vertical and horizontal lines locate the SM point denoted by a star and the best-fit point is denoted by a triangle.
FIG. 6. CPC4:The CL regions of CPC4-A [Left] and CPC4-HP [Right] subfits in two-parameter planes: The contour regions shown are for ∆χ 2 ≤ 2.3 (red), ∆χ 2 ≤ 5.99 (green), ∆χ 2 ≤ 11.83 (blue) above the minimum, which correspond to confidence levels of 68.27%, 95%, and 99.73%, respectively.In each frame, the vertical and horizontal lines locate the SM point denoted by a star and the best-fit point is denoted by a triangle.

FIG. 7 .
FIG. 7. CPC5:The CL regions of the two CPC5 subfits in two-parameter planes: [Left] AHC [Right] LUB.The contour regions shown are for ∆χ 2 ≤ 2.3 (red), ∆χ 2 ≤ 5.99 (green), ∆χ 2 ≤ 11.83 (blue) above the minimum, which correspond to confidence levels of 68.27%, 95%, and 99.73%, respectively.In each frame, the vertical and horizontal lines locate the SM point denoted by a star and the best-fit is denoted by a triangle.
The best-fitted values of the gauge-Higgs couplings C W,Z and the Yukawa couplings of C S u,d,τ are the same at the two generate minima.We observe that, around the SM values of C S τ = C S µ = 1, the 1σ regions of the normalized couplings of C S τ 1σ = [0.876, 0.954] and C S µ 1σ = [0.930, 1.179] marginally overlap with no violation of lepton universality.Comparing the CL regions shown in the left (right) panel of Fig. 8 with those in the upper (middle and lower) frames of the right panel of Fig. 7, we observe that the CL regions CPC6-CSBLUB are very similar to those of CPC5-LUB.

FIG. 10 .
FIG. 10.CPV4 and CPV5: [Left] The CL regions of CPV4-IUF (left and middle) and CPV4-HCC (right) in two-parameter planes.[Right]The CL regions of CPV5-IUHCC in two-parameter planes.The contour regions shown are for ∆χ 2 ≤ 2.3 (red), ∆χ 2 ≤ 5.99 (green), ∆χ 2 ≤ 11.83 (blue) above the minimum, which correspond to confidence levels of 68.27%, 95%, and 99.73%, respectively.In each frame, the vertical and horizontal lines locate the SM point denoted by a star and the best-fit point is denoted by a triangle.

In
CPV7-A, C V is consistent with the SM with the 1σ error less than 4%.The scalar couplings C S u,d,ℓ below the SM with the positive 1σ errors of 4-8%.The negative 1σ errors are larger or much larger: about 10% for C S u and almost 100% for C S d,ℓ .Comparing with the best-fitted values in CPC4-A, we find that the central values and the positive errors of C S u,d,ℓ are similar while the negative errors extend to the negative direction and the positive and negative regions are connected for C S d,ℓ : compare the CL regions shown in the upper frames of the left panel of Fig 6 (CPC4-A) and those in the middle frames of Fig 11 (CPV7 +0.25 −0.19 (95% CL) , (57) which leads to the enhanced Higgs decay into Zγ by the amount of 40% at the upper boundary of the 95% CL region with the larger errors compared to CPC2-I, see the four CPC4-A and CPC6-CSBLUB frames of Fig. 12.We observe that CP violation does not alter the situation, see the two CPV7-A frames of Fig. 12.These observations indicate that one might need nonvanishing ∆S Zγ ∼ −5 to accommodate the measured H → Zγ signal strength of 2.2 ± 0.7 comfortably.For global fits including the H → Zγ data, see Appendix E.

FIG. 12 .
FIG. 12. Predictions for µ(Zγ) in CPC2-I, CPC4-A, CPC6-CSBLUB, and CPV7-A.The upper frames are versus CV , CV , CW , and CV and the lower ones versus C S f , |C S d |, |C S d | and (C S d ) 2 + (C P d ) 2 from CPC2-I to CPV7-A.In each frame, the horizontal line at µ(Zγ) = 1.5 denotes the lower boundary of the shaded 1σ region of the measured H → Zγ signal strength of 2.2 ± 0.7 [14].The contour regions shown are for ∆χ 2 ≤ 2.3 (red), ∆χ 2 ≤ 5.99 (green), ∆χ 2 ≤ 11.83 (blue) above the minimum, which correspond to confidence levels of 68.27%, 95%, and 99.73%, respectively.In each frame, the vertical and horizontal lines locate the SM point denoted by a star and the best-fit point is denoted by a triangle.

FIG. 13 .
FIG.13.Goodness of fit of the CPCn (blue boxes) and CPVn (red triangles) subfits considered in this work.The SM point is denoted by a star.

TABLE III .
(LHC: 13 TeV) ATLAS Run 2 data on signal strengths → γγ decay.For each production-times-decay mode, we note that the measurements of the H → γγ decay in tH production, H → ZZ * in WH, ZH, ttH and tH, H → W W * in WH and ZH, and H → bb in ggF+bbH and VBF are now challenging and they might be significantly improved in near future. H

TABLE VI .
LHC signal strengths for individual Higgs production processes and decay modes obtained from the signal strengths in TableII(Run 1), Table III (Run 2 ATLAS), and Table IV (Run 2 CMS).Note that, being different from the signal strengths in Table V, no assumptions have been imposed on the Higgs boson branching ratios and/or production cross sections.The combined signal strengths are shown in the last line including all the 76 signal strengths available.

TABLE VII .
[49]LO tHq and tHW cross sections in fb at √ s = 13 TeV for the three values of g S H tt taking g P H tt = 0. To calculate the cross sections, we use MG5 aMC@NLO[48]with NN23LO PDF set[49]and no generator-level cuts are applied.MH = 125 GeV and Mt = 172.5 GeV are taken.

TABLE VIII .
The

TABLE X .
CPCn fits and their subfits considered in this work.Varied parameters are denoted by √ in each subfit of CPCn and the SM value of either 0 or 1 is assumed otherwise.

TABLE XI .
CPC1: The best-fitted values in the four CPC1 subfits.Also shown are the corresponding minimal chi-square per degree of freedom (χ 2 min /dof), goodness of fit (gof), and p-value against the SM for compatibility with the SM hypothesis.For the SM, we obtain χ 2 SM /dof = 82.3480/76and gof = 0.2895.

TABLE XII .
CPC2: The best-fitted values in the eight CPC2 subfits.Also shown are the corresponding minimal chi-square per degree of freedom (χ 2 min /dof ), goodness of fit (gof ), and p-value against the SM for compatibility with the SM hypothesis.For the SM, we obtain χ 2 SM /dof = 82.3480/76and gof = 0.2895.
ZZ * ) = 0.97 ± 0.08, see TableVI.The SM point lies outside of the 68% CL region, see the lower-left frame of the left panel of Fig.

TABLE XIII .
CPC3 and CPC4: The best-fitted values in the five CPC3 and two CPC4 subfits.Also shown are the corresponding minimal chi-square per degree of freedom (χ 2 min /dof ), goodness of fit (gof ), and p-value against the SM for compatibility with the SM hypothesis.For the SM, we obtain χ 2 SM /dof = 82.3480/76and gof = 0.2895.

TABLE XIV .
CPC5 and CPC6: The best-fitted values in the two CPC5 and one CPC6 subfits.Also shown are the corresponding minimal chi-square per degree of freedom (χ 2 min /dof ), goodness of fit (gof ), and p-value against the SM for compatibility with the SM hypothesis.For the SM, we obtain χ 2 SM /dof = 82.3480/76and gof = 0.2895.Note that there are two degenerate minima for the positive and negative values of either C S ℓ (CPC5-AHC) or C S µ (CPC5-LUB and CPC6-CSBLUB).

TABLE XVI .
CPV2 and CPV3: The best-fitted values in the four CPV2 and five CPV3 subfits.Also shown are the corresponding minimal chi-square per degree of freedom (χ 2 min /dof ), goodness of fit (gof ), and p-value against the SM for compatibility with the SM hypothesis.For the SM, we obtain χ 2 SM /dof = 82.3480/76and gof = 0.2895.

TABLE XVII .
CPV4, CPV5, and CPV7: The best-fitted values in the CPV4, CPV5, and CPV7 fits.Also shown are the corresponding minimal chi-square per degree of freedom (χ 2 min /dof ), goodness of fit (gof ), and p-value against the SM for compatibility with the SM hypothesis.For the SM, we obtain χ 2 SM /dof = 82.3480/76and gof = 0.2895.

TABLE XVIII .
[66]54] at N 3 LO and σ P ggF at NNLO for several combinations of the relevant Yukawa couplings obtained by using SusHi-1.7.0[53,54]with PDF4LHC15[66].For each combination of the Yukawa couplings, σ bbH at NNLO is also shown. W consider three values of √ s = 7 TeV, 8 TeV, and 13 TeV and MH = 125 GeV has been taken.The renormalization and factorization scales are chosen µR = µF = MH /2 for σggF and µR = 4µF = MH for σ bbH .When g S,P H tt = 1 and g S,P H bb = g S,P H cc = 0, the LO ggF cross sections are also shown in parentheses.
[53,54]eXVIII, we present various ggF and bbH cross sections obtained by using SusHi-1.7.0[53,54]for several combinations of the g S,PH qq couplings at√ s = 7TeV, 8 TeV, and 13 TeV.Neglecting σ S,P cc < ∼ 0.01 pb and σ S,P S H tt ) 2 + 0.007 (g S H bb ) 2 − 0.050 (g S H tt g S H bb ) − 0.010 (g S H tt g S H cc ) + 2.248 (g P H tt ) 2 + 0.007 (g P H bb ) 2 − 0.097 (g P H tt g P H bb ) − 0.018 (g P H tt g P H cc ) .(A.8) Similarly, using the cross sections at √ s = 7 TeV and 8 TeV shown in Table XVIII, we also obtain

TABLE XXII .
Correlations ρxy = ρ (x, y) between the two fitting parameters of x and y in CPC2, CPC3, and CPC4.