Higgcision in the Two-Higgs Doublet Models

We perform global fits to general two-Higgs doublet models (2HDMs) with generalized couplings using the most updated data from ATLAS, CMS, and Tevatron. We include both scenarios with CP-conserving and CP-violating couplings. By relaxing the requirement on the discrete symmetries that are often imposed on the Yukawa couplings, we try to see which of the 2HDMs is preferred. We found that (i) Higgcision in 2HDMs can be performed efficiently by using only 4 parameters including the charged Higgs contributions to the Higgs couplings to two photons, (ii) the differences among various types of 2HDMs are very small with respect to the chi-square fits, (iii) $\tan\beta$ is constrained to be small, (iv) the $p$-values for various fits in 2HDMs are worse than that of the standard model. Finally, we put emphasis on our findings that future precision measurements of the Higgs coupling to the scalar top-quark bilinear ($C_u^S$) and $\tan\beta$ may endow us with the discriminating power among various types of 2HDMs especially when $C_u^S$ deviates from its SM value 1.


I. INTRODUCTION
Upon the observation of a new boson at a mass around 125 GeV at the Large Hadron Collider (LHC) [1,2], the Higgs precision (Higgcision) era has just begun. A study based on a generic framework for the deviations of the couplings from their standard model (SM) values shows [4] that the SM Higgs boson [3] provides the best fit to all the most updated Higgs data from ATLAS [5,6], CMS [7][8][9][10], and Tevatron [11,12].
In addition to a number of more or less model-independent studies , there are also studies done in the 2HDM [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58] and supersymmetric [59][60][61][62][63] frameworks. In this work, we perform global fits to the general 2HDMs (Higgcision in 2HDMs) closely following the generic framework suggested in Ref. [4]. We use the most updated data from the ATLAS, CMS, and the Tevatron and include the scenarios with CP-conserving (CPC) and CP-violating (CPV) couplings. We find that Higgcision in 2HDMs can be performed very efficiently by using only 3 parameters (C S u , C P u , and tan β, as shown later), if one can neglect the charged-Higgs contribution to the Higgs couplings to two photons. To consider the case when the charged-Higgs contribution to the Hγγ couplings is significant, one may need only one additional parameter.
Furthermore, we relax the requirement on the discrete symmetries, which are often imposed on the Yukawa couplings to guarantees the absence of tree-level Flavor Changing Neutral Current (FCNC) [64], to see which of the 2HDMs is preferred. We find that the differences in the chi-squares among various types of 2HDMs are very small and one cannot see any preferences in both the CP-conserving and CP-violating cases.
A number of important findings in this work are: 1. the SM provides the best fit in terms of p-values. The general 2HDM fits at most improve marginally in the total χ 2 at the expense of additional parameters though, and so the p-values do not improve at all; 2. the differences among various types of 2HDMs are negligible in fitting the Higgs data; 3. the gauge boson coupling C v is constrained to be close to 1, which means that the observed Higgs boson is responsible for the most part of the electroweak symmetry breaking; and 4. the tan β is constrained to a small value.
Finally, we emphasize that future precision measurements of C S u and tan β can provide us with the discriminating power among various types of 2HDMs especially when C S u deviates from its SM value 1.
The organization of the paper is as follows. In the next section, we describe the interactions of the Higgs bosons, including deviations in the Yukawa couplings and deviations in the loop functions of Hγγ, Hgg, and HZγ vertices, as well as the notation used in the analysis. In Sec. III, we fix the Higgs potential and Yukawa couplings of the general 2HDMs under consideration and describe how to perform Higgcision in 2HDMs. We articulate that only 4 fitting parameters are needed if we concentrate on the couplings of the candidate for the 125 GeV Higgs boson. We present the results of various fits in Sec. V and conclude in Sec. VI.

II. FORMALISM
For the Higgs couplings to the SM particles assuming the Higgs boson is a generic CPmixed state without carrying any definite CP-parity, we follow the conventions and notations of CPsuperH [65][66][67] in which the Higgs couplings to fermions are given as where f = u, d, l stands for the up-and down-type quarks and charged leptons, respectively, and those to the massive vector bosons are In the SM, g S Hf f = 1, g P Hf f = 0, and g HW W = g HZZ ≡ g HV V = 1. For the loop-induced Higgs couplings to two photons, two gluons and Zγ, and their relevance to the couplings g S,P Hf f and g HV V , we refer to Refs. [4,[65][66][67]. Without loss of generality, we use the following notation for the parameters in the fits: where ∆S γ and ∆P γ denote additional loop contributions to the loop factor S γ and P γ , respectively; and similarly for ∆S g and ∆P g . The ∆Γ tot represents an additional nonstandard decay width of the Higgs boson (e.g., decay into the lighter Higgses). Here we assume generation independence and also custodial symmetry between the W and Z bosons.
Our analysis is based on the theoretical signal strength which may be approximated as the product where P = ggF, VBF, VH, ttH denote the production mechanisms and D = γγ, ZZ, W W, bb, ττ the decay channels. For explicit expressions of µ(P) and µ(D), we again refer to Ref. [4], but by noting they are basically given by the ratios of the Higgs couplings to the corresponding SM ones.
On the other hand, the Yukawa couplings are given in the interactions where We note that there is a freedom to redefine the two linear combinations of Φ 2 and Φ 1 to eliminate the coupling of the up-type quarks to Φ 1 [69]. The 2HDMs are classified according to the values of η l 1,2 and η d 1,2 as in Table I.  By identifying the couplings Here we take the convention with ξ = 0 and the couplings h u,d,l are supposed to be real.
Nevertheless, in order to calculate the signal strengths on which our chi-square analysis is based, we need to know only the couplings of the 125 GeV Higgs boson. Regarding the i-th Higgs boson H i as the candidate for the 125 GeV Higgs boson, and by looking into Eqs. (11) and (12), the relevant Higgs couplings can be fully determined by knowing the components (1) we find where C S u = g S H iū u and C P u = g P H iū u and the orthogonality relation Therefore, by specifying only the 3 parameters of C S u , C P u , and t β , the couplings of the 125 GeV Higgs to all the SM fermions can be determined in each 2HDM as summarized in Table II. In addition, the Higgs coupling to the massive vector bosons is determined by To recapitulate, we need 13 parameters (plus one sign) to fix all the Higgs couplings to the SM particles and the Higgs boson spectrum fully in general 2HDMs. In contrast, only 3 parameters are needed for the couplings of the 125 GeV Higgs candidate to the SM fermions and massive vector bosons. These 3 parameters are the two couplings of the 125 GeV Higgs candidate to the scalar and pseudoscalar top-quark bilinears (C S u and C P u , respectively) and tan β. One may use C v instead of tan β as shown later. In this work, we take advantage of the avenue with the smaller number of parameters to analyze the Higgs data.
With C S u , C P u , and C v (or tan β) given, we also need to know the charged Higgs contribution to the Higgs coupling to two photons in order to calculate the signal strengths. The charged Higgs contribution to the Higgs coupling to two photons is given by where The with g The effective couplings g αH + H − indeed involve all of the Higgs quartic couplings again and read [70] 2 : Therefore, in order to include (∆S γ i ) H ± one may specify all the quartic couplings and the charged Higgs mass in principle, but, then, the situation goes back to the original case with 13 parameters plus one sign. Nevertheless, even in this case one can still keep the spirit of efficiency and simplicity by treating (∆S γ i ) H ± itself as another free parameter in addition to the other three ones C S u , C P u and C v . And then, the results on (∆S γ i ) H ± could be directly interpreted in terms of the coupling g H i H + H − of the 125 GeV Higgs boson to the charged Higgses and the charged Higgs boson mass M H ± , as shown in Eq. (19).
One caveat of our approach to analyze the Higgs data with only 3 or 4 parameters is that one cannot say much about the other two neutral Higgs bosons and the charged one which, in principle, can be either heavier or lighter than the candidate for the 125 GeV Higgs. Before moving to the next section to present the results of various 2-, 3-and 4-parameter fits, we would like to briefly comment on the status of experimental searches for the additional Higgs bosons.
At the LHC, both the ATLAS and CMS collaborations have searched for the additional neutral Higgses bosons up to 1 TeV through their decays into two massive vector bosons, [71,72]. Without observing any positive signal, they put an upper bound on the relevant cross section σ(pp → H i → V V ) 3 . The ATLAS collaboration performed the neutral Higgs-boson searches through the tau-lepton channel, H i → τ τ [73]. While this applies for both the CP-even and CP-odd neutral Higgses up to 500 GeV, it was reported that the constraint for the additional CP-even Higgs from this channel is weaker than that from H i → ZZ [74].
For the charged Higgs boson with mass around a few hundred GeV, the strongest constrain may come from BR(B → X s γ) through the additional loop contributions from the charged-Higgs bosons to the process b → sγ [74]. When the charged Higgs boson is lighter than the top quark, it can be searched at the LHC through the top-quark decay channel t → H + b with the charged Higgs boson subsequently decaying into cb, cs, and τ + ν τ . The direct searches of the charged Higgs boson at the LHC also set limits on the interactions of charged Higgs boson, but their constraints are still weaker than those fromB → X s γ [74]. Higgs data only. This may deserve an independent study and we will discuss these crucial issues in detail in a future publication.

IV. FITS
As shown in the previous section, the whole analysis of the couplings of the observed Higgs boson (denoted by H i ) in 2HDMs, including the CP-conserving and CP-violating cases, can be performed with only 4 parameters: In particular, we consider the following cases with respect to CP-conserving or CP-violating, and with/without charged Higgs contributions: • CP-conserving (CPC) cases Here CPC and CPV represent CP-conserving and CP-violating fits, respectively, and the number denotes the number of varying parameters in each fit. In CPC2 and CPV3, the charged Higgs contribution (∆S γ ) H ± = 0. Note that the varying parameters should satisfy the following relations due to the unitarity of the mixing matrix: One can use tan β in place of C v in the analysis by exploiting the relation derived from Eq. (18): which is independent of sign [O φ 1 i ]. When C S u = 1 and C P u = 0, the above relation becomes s 2 β = (1 + C v )/2, which leads to tan β = ∞ in the SM limit of C v = 1. On the other hand, as in many models beyond the SM, if C S u and/or C P u deviate from its SM values 1 and 0, respectively, one may end up in the opposite limit, tan β = 0, when the dynamics of the fit pushes C v to its maximally allowed value or 1. In practice, one may wish to avoid the regions with small or (very) large tan β to maintain the perturbativity of the top and bottom Yukawa couplings h t and h b , respectively. We therefore restrict the range of tan β between 10 −4 and 10 2 .
Before presenting our numerical results, we briefly review the current Higgs data. Current Higgs data focus on a few decay channels of the Higgs boson: data points in our analysis as in Ref. [4]. To briefly summarize, the chi-square of all these 22 data points relative to the SM is where the numbers in parentheses denote the number of data points in each decay mode.
The chi-square per degree of freedom (dof) is about 18.94/22 = 0.86 and the p-value is about p SM = 0.65. We note the chi-square is dominated by the diphoton data with µ ATLAS ggH+ttH = 1.6 ± 0.4 and µ CMS untagged = 0.78 +0.28 −0.26 . Since the ATLAS data is about 1.5σ larger than the SM while the CMS one is about 1σ smaller, the dynamics of the fit cannot force the parameters to go into either direction.

A. CP conserving fits
In this subsection, we study the CP-conserving case with C P u = 0. In our numerical study, we find that tan β is bounded from above when C S u deviates from its SM value 1. Before presenting numerical results, we look into the correlation among the varying parameters C S u , C v , and tan β.
In the CP-conserving case, Eq. (18) simplifies into with the constraint |s β C S u | ≤ 1, which can be recast into the form taking into account our convention of C v > 0. For a given value of tan β, we find that C v takes the plus(+) sign as C S u increases from −1/t β (where C v = 0) to 1 + 1/t 2 β . While it takes the minus(−) sign when C S u goes from the maximum value 1 + 1/t 2 β back to 1/t β where again C v = 0. Therefore, C v has two positive solutions if C S u lies between 1/t β and 1 + 1/t 2 β . This behavior is shown in the left frame of Fig. 1. From Eq. (24) which now can be rearranged into the form we can see that sin β = 1 or tan β = ∞ along the line C v = C S u . Also, the larger tan β the smaller C v will be. Therefore, tan β will be bounded from above when C v is pushed to be close to 1, unless C S u = 1. To be more precise, we consider the situation in which C v is constrained as C v > (C v ) min .
As illustrated in the right frame of Fig. 1 with three values of C S u = 0.9 (black), 1 (red), and The vertical line shows the location C v = 0.9.
1.1 (blue), we have found that tan β has an upper bound when C S u < (C v ) min for C S u < 1. We observe that the upper bound on tan β is stronger when (C v ) min is closer to 1 but it disappears when (C v ) min < C S u or C S u = 1. On the other hand, when C S u > 1, tan β is always bounded by tan β ≤ 1/ (C S u ) 2 − 1, see Eq. (25). Requiring C v > 0.95, for example, we find tan β < ∼ 6 for C S u = 0.9 and tan β < In the following sub-subsections, we illustrate that the precise and independent measurements of C S u and tan β can tell us the phenomenological viability of 2HDMs and/or enable us to make discrimination among them.
The results for various fits (CPC2, CPC3, CPV3, and CPV4) are tabulated in Tables III and IV, and confidence regions are shown in Figs. 2 -21.

CPC2
The fit CPC2 analyzes the Higgs data by varying C S u and C v (or equivalently log 10 tan β). The total χ 2 , χ 2 /dof, p-value and the best-fit values of C S u , C v , and tan β for the types I -IV of 2HDMs are shown at the top of Table III. We have found that the type-I model gives the smallest χ 2 but the variation of total χ 2 among the 4 types is very small, within 0.29.
Statistically, there is no preference among any type I to IV of 2HDMs. We note that the p-values of the fits are all worse than the SM one p SM = 0.65. The best-fit values for C S u are about 0.9 for type I and III, and about 0.96 for type II and IV. The fitted C v 's are very close to the theoretically allowed maximum value 1 independent of the type. In the actual implementation, we used log 10 tan β as the scanning variable with −4 < log 10 tan β < 2, instead of C v . Again, independent of the type, χ 2 continues to decrease as tan β falls below its lower limit tan β = 10 −4 , though extremely slowly. The best fitted values for tan β are denoted by limit in Table III. We show the contour plots for confidence-level regions as functions C S u vs C v , C S u vs tan β, and C S d vs C S l in Figs. 2 -4, respectively. The regions shown are 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 best-fit point is denoted by the triangle. We note that from Fig. 2 there are two islands and positive C S u is preferred. At 99.7% confidence level (CL), C v > ∼ 0.7. We also find that C S u takes on the values between 0.71 and 1.2 (I), 0.86 and 1.1 (II), 0.71 and 1.2 (III), and 0.86 and 1.1 (IV) at 68.3 % CL. Comparing type I with the other three types, we find that the preference for C S u = 1 is stronger in type II, III, and IV, and C v is more strongly constrained to be close to 1 unless C S u = 1. Furthermore, the tan β = ∞ line with C S u = C v passes through the CL regions only in type I. In Fig. 3, we show the CL regions in the plane of C S u and tan β. For tan β < ∼ 0.5, we find χ 2 is almost independent of tan β for a fixed value of C S u ; while for tan β > ∼ 1, the values of C S u is constrained by C S u ≤ 1 + 1/t 2 β . For type I, as we observed in Fig. 2, the tan β = ∞ line passes through the CL regions and it explains why we can have very large tan β in relatively broader range of C S u . For the other three types it is only possible to have very large tan β in the narrow region around C S u = 1. Thus, in these cases we find that tan β < ∼ 3 (II), 2 (III), 3 (IV) at 99.7 % CL when the best-fit value of C S u is taken in each of the type II, III, and IV. If precise and independent measurements of C S u and tan β are available in future experiments, one can tell the phenomenological viability of 2HDMs. For example, if tan β > ∼ 10 and C S u = 1, then one can rule out the type II, III, and IV models based on Fig. 3.
In Fig. 4, we show the CL regions in the plane of C S d and C S l . From Table II, the following relations C S d = C S l = C S u (I), C S d = C S l (II), C S d = C S u (III), and C S l = C S u (IV) are hold. In Table IV, we can see that the best-fit values of C S d and/or C S l are +1 unless either or both of them are equal to C S u . This can be understood from the relation, for example in type II, with the best-fit values of C S u = 0.963 and tan β = limit = 10 −4 . Note that the positive sign is selected to explain the best-fit values of C S d,l . Taking into account the negative sign, we observe that the points around (C S d , C S l ) = (−1, −1) (II), (C S d , C S l ) = (+1, −1) (III), and (C S d , C S l ) = (−1, +1) (IV) are also allowed at 68.3 % CL even when C S u is positive. So far in this CPC2 fit we only found very small χ 2 differences among the four types.
What if the discrete symmetries are relaxed, do we get a better χ 2 fit? We relax the requirement on the discrete symmetries, which enforces η d, 1,2 to be either 0 or 1, but still require (η d, 1 ) 2 + (η d, 2 ) 2 = 1. We therefore have two more free parameters in our scan, and they are η d, 1 , leading to a four-parameter fit by varying C S u , C v , η d 1 , and η 1 . In Fig. 5, we show the CL regions of the fit by varying C S u , C v , η d 1 , and η 1 in the plane of η d 1 and η 1 4 . We observe that ∆χ 2 < 1 in the whole (η d 1 , η 1 ) plane, and so conclude that one cannot say any preference based on the current Higgs data.

CPC3
In this CPC3 fit, we vary three parameters: C S u , C v (or equivalently log 10 tan β), and (∆S γ ) H ± . The total χ 2 , χ 2 /dof, p-value and the best-fit values of C S u , C v (tan β), and (∆S γ ) H ± for the types I -IV of 2HDMs are shown in the upper half of Table III. We show the contour plots for confidence-level regions as functions C S u vs C v , C S u vs tan β, C S u vs (∆S γ ) H ± , and C S d vs C S l in Figs. 6 -9, respectively. We found that type II gives the smallest χ 2 but the variation of total χ 2 among the four types is very small, within 0.34. The CPC3 fit is slightly better than the CPC2, as it has one more parameter in the fit. However, the p-values of the fits are still worse than the SM one (p SM = 0.65). The best-fit values for C S u are about ±0.92 (I), −0.82 (II), −0.91 (III), and 0.96 (IV) and those of C v are 0.97 for I and III and 1 for II and IV. We also implement independent fits with log 10 tan β as the scanning variable taking −4 < log 10 tan β < 2, instead of C v . The best-fit values for tan β are either small or very small, except for type I with positive C S u . Again, we note that χ 2 hardly changes as tan β varies in wide range of parameter space.
For (∆S γ ) H ± , we have obtained (∆S γ ) H ± −0.8 or 2.3 when C S u ∼ +0.9 or −0.9, respectively. This can be understood from the numerical expression for S γ [4] When C S u changes from +0.9 to −0.9, (∆S γ ) H ± changes from −0.8 to +2.3 so that the sum The contour plots for the CL regions in the plane of C S u vs C v for type I -IV are shown in Fig. 6, which can be directly compared to Fig. 2. In contrast, the negative C S u is now equally as good as the positive one. We show the CL regions in the plane of C S u and tan β in Fig. 7. For the negative C S u case, we find that tan β is smaller than ∼ 0.6 at 99.7 % CL. In Fig. 8, we show the CL regions in the plane of C S u and (∆S γ ) H ± . For positive C S u , it lies between 2 and −4 while (∆S γ ) H ± > −0.7 ∼ −1.6 for negative C S u at 99.7 % CL. The CL regions for C S l and C S d are similar to the CPC2 case as shown in Fig. 9 but with the larger regions allowed at 68.5 % CL around the negative values of couplings.
The single parameter (∆S γ ) H ± can be interpreted in terms of the charged Higgs mass M H ± and the neutral Higgs coupling to the charged Higgses g H i H + H − , as in Eq. (19). In Fig. 10, we show the CL regions in the plane of M H ± vs g H i H + H − . Since the variation of χ 2 is very mild, we add one more region with ∆χ 2 ≤ 1 (black). The thick cyan lines denote the points giving the best-fit values of (∆S γ ) H ± in each type given by Eq. (19). We see that a smaller charged Higgs mass is preferred when g H i H + H − < 0, because this corresponds to C S u < 0 and so a larger (∆S γ ) H ± ≈ 2.3 is required. If the charged Higgs mass is larger than ∼ 300 GeV as in the type II model constrained by B(b → sγ), we can see that the positive C S u case with (∆S γ ) H ± ∼ −0.8 is somewhat preferred. Nevertheless, the variation of χ 2 is not large enough to have a conclusive statement based on the current Higgs data.

B. CP violating fits
In this subsection, we study the CP-violating case with a nonzero C P u in addition to C S u , C v (or, equivalently, tan β), and (∆S γ ) H ± . In our numerical study, we again find that tan β is bounded from above when C S u deviates from its SM value 1. So, as in the CP-conserving case, the precise and independent future measurements of C S u and tan β can tell us the phenomenological viability of 2HDMs, thus providing some possible model discriminating power.

CPV3
In the CPV3 fit, we vary C S u , C P u , and C v (or equivalently log 10 tan β). The other couplings C S,P d,l are given by the relations shown in Table II. The total χ 2 , χ 2 /dof, p-value, and the best-fit values for C S u , C P u , and C v (tan β) for the types I -IV 2HDMs are shown in the lower half of Table III. We show the contour plots for confidence-level regions as functions C S u vs C P u , C S u vs C v , C S u vs tan β, C S d vs C P d , and C S l vs C P l in Figs. 11 -15, respectively. We found that type II gives the smallest χ 2 and the variation of total χ 2 among the 4 types is within 1.2, which is about 4 times larger compared to the CP-conserving case.
Yet, such small χ 2 differences cannot help us to preferentially select one of the types. The best p-value for type II is 0.578, which is the largest among all the fits considered in this work, but it is still smaller than the SM p SM = 0.65. The CL regions in the C S u and C P u plane are shown in Fig. 11. A positive C S u is in general preferred and it takes a value between 0. 44  . We note that maximal CP violation with C S u ∼ |C P u | is possible even when C v 1. This can be understood by considering the relation Eq. (18), which takes on a form of in the tan β = 0 limit. Taking an example of C S u = C P u = 1/2, one may have up to O(β 3 ). Hence, although the 126-GeV observed state is mostly CP-even dominated by the φ 1 component, it can have maximally CP-violating couplings to the up-type quarks with C S u = |C P u | = 1/2. In Figs. 12 and 13, we show the CL regions in the C S u vs C v and C S u vs tan β planes, respectively. Compared to the CPC case, we observe that the two islands are now merged together, except for type III. We again find that tan β is bounded from above: tan β < ∼ 1 (II), tan β < ∼ 3 (III), and tan β < ∼ 2 (IV). As in the CPC case, considerable deviation of C S u from 1 for large tan β > ∼ 10 is not possible in the type II, III, IV models. In Fig. 14, we show the Higgs couplings to the down-type quarks. The behavior can be understood by observing the relations C S d = C S u and C P d = −C P u (I and III) and β /s 2 β (C P u ) 2 1/2 and C P d = t 2 β C P u (II and IV), see Table II. Note that |C P d | < ∼ 1 at 99.7 % CL. We observe large CP violation is possible in the Higgs couplings to the down-type quarks.
In Fig. 15, we show the Higgs couplings to the charged leptons. Now the couplings are given by C S l = C S u and C P l = −C P u (I and IV) and C S l = ± 1 + t 2 β [1 − (C S u ) 2 ] − t 4 β /s 2 β (C P u ) 2 1/2 and C P l = t 2 β C P u (II and III). Again we note that |C P l | < ∼ 1 at 99.7 % CL and large CP violation is also possible in the Higgs couplings to the charged leptons.
Before we close this sub-subsection, we make a comment on the figures for the CL regions in the planes of C S d vs C P d and C S l vs C P l . Unless (C S d,l , C P d,l ) = (C S u , −C P u ), the boundaries of the CL regions are somewhat fuzzy as shown in the frames for type II and IV of Fig. 14 and in those for type II and III of Fig. 15. We figure out that this is because one has (C S d,l , C P d,l ) ∼ (1, 0) in most of the parameters space due to the coupling relations shown in Table II. Furthermore, we have the fewer points on the negative side of C S d or C S l . For the couplings C S d,l to be negative, the negative sign needs to be chosen for O φ 1 i in Eq. (17). But we note that the other positive sign is chosen mostly for O φ 1 i due to the choice of C v > 0 made in the analysis. Similar behavior happens in Fig. 20 and Fig. 21.

CPV4
In the CPV4 fit, we vary C S u , C P u , C v (or log 10 tan β equivalently), and (∆S γ ) H ± . The other couplings C S,P d,l are given by the relations shown in Table II. The total χ 2 , χ 2 /dof, p-value and the best-fit values for C S u , C P u , C v (tan β), and (∆S γ ) H ± for the four types of 2HDMs can be found in the lower half of Table III. We show the contour plots for confidencelevel regions as functions C S u vs C P u , C S u vs C v , C S u vs tan β, C S u vs (∆S γ ) H ± , C S d vs C P d , and C S l vs C P l in Figs. 16 -21, respectively. We find that type II gives the smallest χ 2 and its variation among the 4 types is within 0.57, which is smaller than that of the CPV3 fits.
The p-values of the CPV4 fits are also worse than the CPV3 fits. In type II, we note the best-fit value for C S u is almost 0 and those of C P u are very small except for type II. Therefore, in terms of the best-fit values the measure of CP-violating effect 2C is not significant in all 4 types of 2HDMs. Nevertheless, the CP violation could be significant taking account of the errors. For the Higgs couplings to the down-type quarks and charged leptons, we find that all the couplings C P d and C P l are almost vanishing: see Table IV. The best-fit values for C v are about 0.97 (I and III) and 1 (II and IV) and those for tan β are O(0.1), except for type I with positive C S u , where the best-fit value is 6.5. As will be shown below in the figures, variation of χ 2 vs of tan β is small in a large region of parameter space. This also can be understood from Eq. (27).
In Fig. 16, we show the CL regions in the C S u and C P u plane, and note that the positive and negative C S u regions are providing equally good fits. The 68 % CL regions of C S u are: . Also, C P u varies between ±0.5 (I, III) and ±0.7 (II, IV) in the 68 % CL regions. Therefore, the maximal CP violation with |C S u | = |C P u | is still possible.
We show the CL regions in the C S u -C v and C S u -tan β planes in Figs. 17 and 18, respectively. C v > ∼ 0.8 at 68 % CL and tan β are bounded from above for the type II, III, and IV, except for a narrow region around C S u = 1. The CL regions in the plane of C S u and (∆S γ ) H ± are shown in Fig. 19

V. DISCUSSION
In this work, we have applied our previous model-independent approach [4], which analyzes all the observed Higgs boson signal strengths and fits to all the Higgs boson couplings, to the 2HDMs. In 2HDMs, the Higgs couplings to up-type and down-type quarks, and charged leptons are related by a set of relations shown in Table II. We have shown that the whole analysis can be performed with at most 3 independent parameters: C S u , C v (or tan β), and (∆S γ ) H ± for CP-conserving scenarios, and only one more parameter C P u for the CP-violating scenarios. A number of relationships among the couplings of the up-and down-type quarks and charged leptons have been derived such that we need only C S u and C P u . A set of discrete symmetries are often imposed in literature in order to eliminate flavorchanging neutral currents, denoted by the parameters η d,l 1,2 , which take up the values either 0 or 1. The four combinations of (η d 1 , η l 1 ) = (0, 0), (1, 1), (0, 1), (1, 0) correspond to type I, II, III, and IV, respectively. We have demonstrated that the current Higgs boson data have no preference for any of the four types of 2HDMs, because statistically the χ 2 difference among type I-IV is only 0.3 for CPC cases and 1.2 for CPV cases: see Table III. We also relaxed the discrete symmetries to allow continuous values for η d,l 1,2 subject to normalization (η d,l 1 ) 2 + (η d,l 2 ) 2 = 1, and we found that in the whole plane of 0 ≤ η d 1 , η l 1 ≤ 1 the χ 2 differences among the best-fits are all within χ 2 < 1.2. It is one of the main findings in this workno particular preference among type I to IV as long as the current Higgs boson data are concerned.
The Higgs data used are almost the final set out of the 7 TeV and 8 TeV runs at the LHC. Further improvement to the fits will only be possible when more data are pouring in the next run of 2015. So far, the data have pointed to the SM Higgs boson with a large p-value, while all other extensions to the SM, such as the 2HDMs studied in this work or more model-independently in Ref. [4], provide fits with smaller p-values than the SM. It means that the SM Higgs boson is currently the best explanation to all the Higgs boson data.
We offer a few more comments before we conclude.
1. The up-type and down-type (charged lepton) Yukawa couplings are related by quark masses and tan β. Therefore, one set of parameters C S u , C P u , and tan β is sufficient to define all the fermionic couplings.
2. When we relax the discrete symmetries by varying η d 1 and η l 1 , we found the best-fit values for them are neither 0 nor 1. However, the χ 2 differences in the whole plane of η d 1 vs η l 1 are too small to claim any preference statistically. 5. Future precision measurements of C S u and tan β can provide us with the discriminating power among various types of 2HDMs especially when C S u deviates from its SM value 1.     11. The confidence-level regions of the fit by varying C S u , C P u , and log 10 tan β (CPV3 case) in the plane of C S u vs C P u for Type I -IV. The contour regions shown are 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 best-fit points are denoted by the triangle.
FIG. 15. The same as in Fig. 11 but in the plane of C S l vs C P l for Type I -IV (CPV3). The description of the confidence regions is the same as Fig. 11.