New Emerging Results in Higgs Precision Analysis Updates 2018 after Establishment of Third-Generation Yukawa Couplings

We perform global fits of the Higgs boson couplings to all the 7 TeV, 8 TeV, and 13 TeV data available up to the Summer 2018. New measurements at 13 TeV extend to include the Higgs signal strengths exclusively measured in associated Higgs production with top-quark pair and the third-generation Yukawa couplings now have been established. Some important consequences emerge from the global fits. (i) The overall average signal strength of the Higgs boson stands at $2\sigma$ above the SM value ($\mu = 1.10 \pm 0.05 $). (ii) For the first time the bottom-quark Yukawa coupling shows a preference of the positive sign to the negative one. (iii) The negative top-quark Yukawa coupling is completely ruled out unless there exist additional particles running in the $H$-$\gamma$-$\gamma$ loop with contributions equal to two times the SM top-quark contribution within about 10 \%. (iv) The branching ratio for nonstandard decays of the Higgs boson is now below 8.4\% at the 95\% confidence level.


I. INTRODUCTION
Ever since the discovery of a standard model (SM) like Higgs boson in 2012 [1,2], the main focus of the LHC experiments has been put on fully establishing its identity. Though the initial data sets till the summer 2013 indicated that it might be different from the SM Higgs boson [3], the data sets collected till the summer 2014 showed that the data is best described by the SM Higgs boson [4]. Ever since then more production channels and decay channels of the Higgs boson are established. On the production side, in addition to gluon fusion (ggF), vector-boson fusion (VBF), the associated production with a V = W/Z boson (VH), and the associated production with a top-quark pair (ttH) have been extensively investigated [5,6]. On the decay side, H → bb [7,8] and H → τ τ [9,10] were also very recently established. It is the right timing to perform the global fits to all Higgs-boson signal strengths in various scenarios of new physics, generically labeled by CPCn and CPVn in this work with n standing for the number of fitting parameters.
Some very interesting results emerge from the new global fits, which were not realized previously.
1. The combined average signal strength of the Higgs boson now stands at a 2-σ deviation from the SM value, namely µ exp = 1.10 ± 0.05.
2. For the first time the bottom-Yukawa coupling shows statistical difference between the positive and negative signs. Thanks to the discriminating power of the Higgs-gluon vertex S g the positive sign of the bottom-Yukawa is more preferred than the negative one.
3. Previously in 2014 the fits still allowed the negative sign of the top-Yukawa coupling at the 95% confidence level (CL). Now with more precisely measured signal strengths together with the establishment of the associated production with the top-quark pair, the negative island of the top-Yukawa is now entirely ruled out, except in the scenarios with non-zero ∆S γ . Even with ∆S γ = 0, it has to be adjusted within 10 % of two times the SM top-quark contribution. This tuning is going to be more and more severe as more data accumulate. 4. The nonstandard (or invisible decay) branching ratio of the Higgs boson is now reduced to less than 8.4% at the 95% CL which improves substantially from the previous value of 19%. This is obtained by varying only ∆Γ tot .
The organization of the paper is as follows. In the next section, we describe briefly our formalism to make this work more self-contained. In Sec. III, we show the data for the Higgs signal strengths. In Sec. IV, we show the results for all the fits. We conclude in Sec.
V. In appendix, we list all the Higgs boson data that we use in our global fitting.

II. FORMALISM
In order to make the current presentation more self-contained, we include here brief description of the formalism that we use in calculating the signal strengths and chi-squares.
We follow the conventions and notations of CPsuperH [11][12][13] for the Higgs couplings to the SM particles assuming the Higgs boson is a generally CP-mixed state without carrying any definite CP-parity.
• Higgs couplings to fermions: Hf g S Hf f + ig P Hf f γ 5 f .
For the SM couplings, g S Hf f = 1 and g P Hf f = 0.
• Higgs couplings to the massive vector bosons: For the SM couplings, we have g HW W = g HZZ ≡ g HV V = 1, respecting the custodial symmetry.
• Higgs couplings to two photons: The amplitude for the decay process H → γγ can be written as where k 1,2 are the momenta of the two photons and 1,2 the wave vectors of the cor- Including some additional loop contributions from new particles, the scalar and pseudoscalar form factors, retaining only the dominant loop contributions from the thirdgeneration fermions and W ± , are given by 1 where τ x = M 2 H /4m 2 x , N C = 3 for quarks and N C = 1 for taus, respectively. The additional contributions ∆S γ and ∆P γ are assumed to be real in our work, as there are unlikely any new charged particles lighter than M H /2.
• Higgs couplings to two gluons: Similar to H → γγ, the amplitude for the decay process H → gg can be written as where a and b (a, b = 1 to 8) are indices of the eight SU (3) generators in the adjoint representation. The decay rate of H → gg is proportional to |S g | 2 + |P g | 2 . Again, including some additional loop contributions from new particles, the scalar and pseudoscalar form factors are given by 1 For the loop functions of F sf,pf,1 (τ ), we refer to, for example, Ref. [11].
The additional contributions ∆S g and ∆P g are assumed to be real again.
• Higgs couplings to Z and γ: The amplitude for the decay process can be written as where k 1,2 are the momenta of the Z boson and the photon (we note that 2k 1 · k 2 = M 2 H −M 2 Z ), 1,2 are their polarization vectors. The scalar and pseudoscalar form factors can be found in Ref. [3].
Finally, we define the ratios of the effective Higgs couplings to gg, γγ, and Zγ relative to the SM ones as follows: Note that the ratios of decay rates relative to the SM are given by |C g | 2 , |C γ | 2 , and |C Zγ | 2 , respectively.
The theoretical signal strength may be written as the product where P = ggF, VBF, VH, ttH denote the production mechanisms and D = γγ, ZZ, W W, bb, ττ the decay channels. More explicitly, we are taking and with Note that we introduce an arbitrary non-SM contribution ∆Γ tot to the total decay width. Incidentally, Γ tot (H) becomes the SM total decay width when g S Hf f = 1, g P Hf f = 0, g HW W,HZZ = 1, ∆S γ,g,Zγ = ∆P γ,g,Zγ = 0.
The experimentally observed signal strengths should be compared to the theoretical ones summed over all production mechanisms: where Q denote the experimentally defined channel involved with the decay D and the decomposition coefficients C QP may depend on the relative Higgs production cross sections for a given Higgs-boson mass, experimental cuts, etc.
The χ 2 associated with an uncorrelated observable is where σ EXP (Q, D) denotes the experimental error. For n correlated observables, we use where V is a n × n covariance matrix whose (i, j) component is given by with ρ denoting the relevant n×n correlation matrix. Note ρ ij = ρ ji , ρ ii = 1, and if ρ ij = δ ij , χ 2 n reduces to i.e., the sum of χ 2 of each uncorrelated observable.

III. HIGGS SIGNAL STRENGTH DATA
In our work, we use the direct Higgs data collected at the Tevatron and the LHC. We use 3 signal strengths measured at the Tevatron. The Higgs-boson data at 7 and 8 (7+8) TeV used in this analysis are the signal strengths obtained from a combined ATLAS and CMS analysis [14]. We also take into account the correlation matrix given in Fig. 27 of Ref. [14].
On the other hand, the 13 TeV data are still given separately by ATLAS and CMS and in different production and decay channels. In total, the number of signal strengths considered is 3 (1.96 TeV) + 20 (7 + 8 TeV) + 41 (13 TeV) = 64. In Appendix, we collect the data sets at the Tevatron and at 7+8 TeV and 13 TeV.
Except for the combined average signal strengths at 7+8 TeV given in Ref. [14], we combine signal strengths of various channels at 1.96 TeV and 13 TeV using a simple χ 2 method and assuming each is Gaussian distributed. We first show the combined ATLAS and CMS signal strengths at 13 TeV strengths for each production and decay channel in  Table I. Before we go to the global fits, we would like to point out a few peculiar features in the data sets, and the average signal strengths.
1. The combined overall signal strength at 7+8 TeV is µ 7+8TeV = 1.09 +0.11 −0.10 [14], which is larger than the SM value by slightly less than 1σ. Table I, it is clear that all decay channels show slight excess over the SM value of 1.0, especially the H → γγ and H → W W * channels. Table I, almost all production modes, except for ttH, show excess above the SM, especially the gluon fusion (ggF). 4. The 13 TeV data shows similar deviations in both ATLAS and CMS results: µ ATLAS 13TeV = 1.09 ± 0.08 and µ CMS 13TeV = 1.1 +0.09 −0.08 . By combining these two results we obtain

Again from
which is about 1.67σ above the SM.
5. Finally, we combine all the signal strengths for the Tevatron at 1.96 TeV, and for 7 + 8 and 13 TeV ATLAS and CMS, and thus obtain µ All = 1.10 ± 0.05 which indicates a 2σ deviation from the SM value.
We summarize the results in Table II.

IV. GLOBAL FITS
We perform global fits in which one or more parameters are varied. They are categorized into CP-conserving (CPC) and CP-violating (CPV) fits, because the current data still allows the observed Higgs boson to be a mixture of CP-even and CP-odd states. Assuming generation independence for the normalized Yukawa couplings of g S,P Hf f , we use the following notation for the parameters in the fits: In most of the fits, we keep the custodial symmetry between the W and Z bosons by taking which is motivated by the data.

A. CP Conserving fits
In CP-conserving fits, we are varying C S u , C S d , C S ,C v;w,z , ∆S g , ∆S γ , and ∆Γ tot while taking C P u = C P d = C P = ∆P γ = ∆P g = 0. All the CP-conserving fits considered in this work are listed here: • CPC2: vary ∆S γ and ∆S g while keeping C S u = C S d = C S = C v = 1 and ∆Γ tot = 0.
• CPCX4: vary C S u , C w , C z , ∆S g while keeping C S d = C S = 1 and ∆S γ = ∆Γ tot = 0.
Note that CPC1 to CPC6 were those originally in our first Higgcision paper [3] while CPCN2 to CPCN4 were those studied in our 2014 update paper [4]. The CPCX2 to CPCX4 are new in this work. The reason why we study more scenarios here is because we want to fully understand the effects of having ∆S g alone, in order to discriminate the contribution from the bottom-Yukawa coupling to Higgs production. In doing so we find that the effect of the bottom-Yukawa coupling becomes sizable in the Higgs-gluon-gluon vertex: numerically flipping the sign of bottom-Yukawa coupling can cause more than 10% change in |S g | while it is less than 0.5% in |S γ |. 1. CPC1 to CPC6 The fitting results for CPC1 to CPC6 are shown in Table III. The corresponding figures for confidence regions are depicted in Fig. 1 to Fig. 5. In the following, we are going through each fit one by one.
In CPC1, the best-fit value for ∆Γ tot is which is 1.6 σ below zero. The p-value of this fit is 0.851, which is indeed better than the SM (p-value = 0.814). This finding is consistent with the average signal strength µ All = 1.10 ± 0.05. Nevertheless, we do not recall any new physics models that reduce the total decay width. From the fit we can determine the upper limit for ∆Γ tot . The 95% CL allowed range for ∆Γ tot = −0.285 +0.38 −0.32 , as shown in Fig. 1. Assuming the fit is consistent with the SM, the 95% CL upper limit for ∆Γ tot = 0.38 MeV (we simply take the central value equal to zero and use the upper error as the upper limit), which translates to a branching ratio B(H → nonstandard) < 8.4% , which improves significantly from the previous value of 19% [4]. In CPC2, we vary ∆S g and ∆S γ -the vertex factors for Hgg and Hγγ, respectively. This scenario accounts for additional charged particles running in the loop of Hγγ vertex and additional colored particles running in the loop of Hgg vertex. The best-fit point (∆S γ , ∆S g ) = (−0.226, 0.016) shows an increase of 3.4% and 2.4% in |S γ | and |S g |, respectively. We note that the error of ∆S g is now ±0.025, which is numerically smaller than the SM bottom-quark contribution of −0.037 to the real part of S g , see Eq. (8), alerting that we have reached the sensitivity to probe the sign of the bottom-quark Yukawa coupling in gluon fusion. The p-value of the best-fit point is about as the SM one. In Fig. 2, we show the confidence-level regions of the fit for ∆χ 2 ≤ 2.3 (red), 5.99 (green), and 11.83 (blue) above the minimum, which correspond to confidence levels of 68.3%, 95%, and 99.7%, respectively.
The corresponding regions for (C γ , C g ) are also shown in the right panel.
In CPC3, ∆Γ tot , ∆S g , and ∆S γ are the varying parameters. The best-fit point shows that the data prefer modification of ∆Γ tot to accommodate the data rather than the other two parameters. It implies that the excesses are seen in most channels, not just the diphoton channel. Nevertheless, the p-value of this fit is very similar to CPC2 and the SM. On the other hand, the better p-value of CPC1 indicates that the data prefer enhancement in all channels, instead of a particular one. The confidence-level regions of the fit are shown in  The CPC4 fit allows C v , C S u , C S d , C S to vary, and it shows two most dramatic changes from previous results [3,4]. (i) The "island" on the negative of C S u in the (C S u , C v ) plane completely disappears, shown in the left panels of Fig. 4. (ii) The middle panels show that C S d now prefers the positive sign to the negative one. It is more clear from the middle-lower panel that the point C S d = −1 has ∆χ 2 > 2 above the minimum at C S d = +1. This is the first time that the data prefer positive bottom-Yukawa coupling to the negative one. The key observation here is that when we change the sign of bottom-Yukawa coupling, the vertex factor S γ only changes by 0.03/6.64 = 0.0045, which is too small compared with experimental uncertainty. On the other hand, the vertex factor S g changes by 0.074/0.651 = 0.11, which now becomes comparable to experimental uncertainty. This is the reason why the positive bottom-Yukawa is more preferred in the scenario with ∆S g = 0. Yet, the current data precision still do not show any preference for the sign of tau-Yukawa coupling, as shown in  Table III, we only show the minimum at C S u,d, ∼ 1 and ∆S γ,g ∼ 0. A substantial improvement from previous results is that the confidence-level regions shown in Fig. 5 are now well separated islands, while in previous results [4] those islands are "connected". For example, in the plane of (C S u , C v ), the negative and positive islands of C S u are now separated but they were connected in previous results. It means that previously C S u = 0 was allowed but not in the current data.
Before moving to CPCN fits, we note that the negative top-quark Yukawa coupling is allowed only in the presence of non-zero ∆S γ which can offset the flipped top-quark contribution to S γ . The required tuning is now δ(∆S γ ) ±0.4 at 1 σ level, which is about 10% of the change in ∆S γ due to the negative top-quark Yukawa coupling. The tuning will be more and more severe as more data accumulate.

CPCN2 to CPCN4
We can see in CPC2 to CPC6, the bottom-Yukawa and tau-Yukawa couplings are not very sensitive to the overall fits, though the bottom-Yukawa shows slight preference on the positive side in CPC4. Here we attempt to use the more effective parameters in the fits.
The best-fits points and their p-values are shown in Table IV, and their corresponding figures in Fig. 6 to 8.
In CPCN2, we vary only C v and C S u . This fit offers a slightly better p-value than the SM. While in CPCN3, we also vary ∆S γ in addition to C v and C S u . We find that it has little improvement over the CPCN2 in terms of total χ 2 but, with one less degree of freedom, the p-value indeed decreases. As shown in Fig. 7, there are two minima: the minimum near (C S u , ∆S γ ) = (1, 0) provides a better solution by ∆χ 2 ≈ 2 than the other one near (C S u , ∆S γ ) = (−1, 3.2). The ∆S γ can compensate the flip in sign of C S u in the vertex factor S γ . However, when C S u flips the sign, |S g | increases by more than 10% leading to a worse After ICHEP 2018  fit. In CPCN4, we vary the four most efficient fitting parameters C v , C S u , ∆S γ , and ∆S g . Therefore, in contrast to CPCN3, the ∆S g here can compensate the sign change in C S u , such that there are four minima in this fit with the same p-value, as shown in Table IV and

CPCX2 to CPCX4
We perform some more fits, which were not considered in our previous works. The bestfit points for CPCX2 to CPCX4 are shown in Table V and the corresponding figures in Fig. 9 to Fig. 11.
The CPCX2 fit involves C v and ∆Γ tot . Both parameters shift from the corresponding SM values in order to enhance the signal strengths. The confidence-level regions are shown in Fig. 9.
In addition to C v and C S u (similar to CPCN2), the CPCX3 fit also varies ∆S g . The result is very similar to CPCN2, but ∆S g has two solutions with the same p-values: see In the CPCX4 fit, we relax the requirement of C w = C z because we can see from the 13 TeV data that the signal strengths for H → W W * are generically larger than those for H → ZZ * , see Table I. The result is shown in Table V. The best-fitted values for C w and C z are within 1σ and C w > C z as demanded by the data. Again there are two solutions for ∆S g : see Fig. 11. We note that, compared to C z which is only constrained by H → ZZ * decay, C w is constrained by both VBF and WH production as well as H → γγ and H → W W * decays. This leads to the narrower ∆χ 2 curve in C w than in C z , as shown in lower frames of Fig. 11.

B. CP Violating fits
For the CP-violating fits, we consider the following 4 scenarios: • CPV2: vary C S u , C P u .
• CPV3: vary C S u , C P u , C v .
The current Higgs boson data ruled out a pure pseudoscalar [15,16], but the data cannot rule out a mixed state [17]. Noting that the CP-odd coupling to gauge bosons only arises from loop corrections, we only allow the top-quark Yukawa coupling and the vertex factors for Hgg and Hγγ to develop sizeable CP-odd couplings. Therefore, CP violation is signaled After ICHEP 2018 by the simultaneous existence of C S u and C P u as in CPV2, CPV3, and CPVN3 or of ∆S γ ,g and ∆P γ ,g in CPV4. The results for CPV2 to CPVN3 are shown in Table VI and the corresponding figures in Fig. 12 to Fig. 15.
The simplest choice CPV2 happens in the coexistence of CP-even and CP-odd top-Yukawa couplings: C S u and C P u . Since the signal strengths are CP-even quantities, in general, they do not contain any CP-odd products of C S u × C P u and S g,γ × P g,γ even though the products are non-vanishing. This is why the confidence-level regions appear like a circle or an arc of a circle in the planes of (C S u , C P u ), (∆S γ , ∆P γ ), and (∆S g , ∆P g ). In Fig. 12, we show two best-fit points with equal p-value for CPV2, indeed the arc joining these two points essentially has the same p-value.
We vary C S u , C P u , and C v in CPV3 fit. The confidence-level regions, shown in Fig. 13, The color code is the same as in Fig. 2. shrink a lot from previous results [4]. Previously, the blue region forms a closed ellipse, but now all regions hardly form a closed ellipse, showing the data are getting much more stringent than before.
We vary ∆S g , ∆S γ , ∆P g , and ∆P γ in CPV4. As explained in our previous work [3], the solutions to ∆S g and ∆P g , as well as to ∆S γ and ∆P γ appear to be an ellipse. It is quite clear in Fig. 14 In the CPVN3 fit, we try a different combination of parameters: C S u , C P u , and ∆Γ tot . With the help of ∆Γ tot the "banana" shaped regions originally in CPV2 now become fattened.

C. Predictions for H → Zγ
Before we close this section, we examine how large C Zγ can be in the scenarios with ∆S γ = 0, assuming the absence of additional particles running in the H-γ-Z loop. The results are shown in Fig. 16. We observe that C Zγ can be as large as 1.2 which may imply

V. CONCLUSIONS
We have performed global fits to the Higgs couplings to gauge bosons and fermions, using all the data from the Tevatron, 7+8 TeV and 13 TeV data from ATLAS and CMS. Overall, the allowed parameter space regions shrink substantially from those in 2014. Notably, the 5. The nonstandard (or invisible decay) branching ratio of the Higgs boson is now reduced to less than 8.4%, which improves substantially from the previous value of 19%. This is obtained by varying only ∆Γ tot . It would be relaxed if more parameters are allowed to vary in the fit.
6. When we relax the custodial symmetry requirement (C w not necessarily equal to C z ), we find that the coupling C w is larger than C z though still within 1σ, and more constrained than C z . 7. We have also made the predictions for H → Zγ by showing the effective coupling C Zγ . In most scenarios, it is predicted to be SM-like. The most extreme allowed value would C Zγ 1.2, which gives a branching ratio 40% larger than the SM value.

APPENDIX
In this appendix, we list all the details of Higgs signal strengths used in our global fitting.