First Glimpses at Higgs' face

The 8 TeV LHC Higgs search data just released indicates the existence of a scalar resonance with mass ~ 125 GeV. We examine the implications of the data reported by ATLAS, CMS and the Tevatron collaborations on understanding the properties of this scalar by performing joint fits on its couplings to other Standard Model particles. We discuss and characterize to what degree this resonance has the properties of the Standard Model (SM) Higgs, and consider what implications can be extracted for New Physics in a (mostly) model-independent fashion. We find that, if the Higgs couplings to fermions and weak vector bosons are allowed to differ from their standard values, the SM is ~ 2 sigma from the best fit point to current data. Fitting to a possible invisible decay branching ratio, we find BR_{inv} = 0.05\pm 0.32\ (95% C.L.) We also discuss and develop some ways of using the data in order to bound or rule out models which modify significantly the properties of this scalar resonance and apply these techniques to the global current data set.


I. INTRODUCTION
Particle physics entered a new era with the announcement of the discovery of a new boson [1] based on excess events in several Higgs search channels using 7 + 8 TeV LHC data collected in 2011-2012. In light of this discovery, it has become obvious that the question of central importance to address now is -what are the properties of the scalar field responsible for the observed excesses?
The answer to this question determines if this field corresponds to the Standard Model (SM) Higgs, with the specific SM mechanism of elegantly breaking electroweak (EW) symmetry, or whether a more complicated mechanism is involved in EW symmetry breaking. We study this question in detail in this paper, characterizing to what degree a SM Higgs is consistent with the current global dataset and presenting several results on the properties of the scalar field. Besides updating arXiv:1207.1717v2 [hep-ph] 21 Aug 2012 and expanding our past results [2], we also present new analyses that emphasize the power of the growing dataset to bound and rule out alternative models or to give hints of New Physics (NP).
It is worth emphasizing that it is very important to specify (and justify) a coherent theoretical framework in which to study the emerging evidence for the scalar field. However, without knowing the ultraviolet (UV) origin of this field, we do not know what effective field theory (EFT), or complete model, should be used to fit the data. We emphasize that at this time, the existing experimental evidence is not sufficiently strong to directly assume that the scalar resonance is the SM Higgs boson, ascribing any deviations in the measured properties of the scalar field directly to the effects of NP interactions expressed through higher dimensional operators. Although this is certainly one possible interpretation of the data (and we will examine the implications of Higgs data for NP in this framework), we emphasize that, in general, one should not assume what one wishes to prove.
Nevertheless, in formulating a theoretical framework for this study, a wealth of other experimental results that are also sensitive to the properties of scalar fields at the weak scale can be distilled into simple physical guiding principles. These are, namely, approximate Minimal Flavour Violation (MFV) [3][4][5][6][7]; respecting the soft Higgs theorems of Refs. [8,9] (i.e. the scalar couples to the SM fields in proportion to their masses); and an effective breaking of custodial symmetry, SU(2) c , [10][11][12] approximately as in the SM. Directly incorporating these principles in the formulation of the effective Lagrangian allows us to restrict our attention to a few simple cases. In order to establish experimentally the properties of the scalar resonance in a model-independent way, one can utilize the effective field theory of the chiral EW Lagrangian coupled to a scalar field that was emphasized in Refs. [2,13] to study recent Higgs signal-strength data. 1 Depending on the assumptions of the UV origin of such a Lagrangian, one is lead to various sets of free parameters to fit the data when studying the consistency of the SM Higgs hypothesis with the current dataset.
We discuss and utilize this framework extensively in this paper to examine the properties of the scalar field emerging from the data.
We also emphasize that with the discovery of a new scalar resonance, one can also use the signal strength properties of the scalar field to bound and rule out models that provide too few signal events as well as models that provide too many signal events. Further, one can also exclude allowed parameter space due to the degree of tension within the dataset that depends on the properties of the scalar field. As the dataset evolves, these techniques become complementary to direct χ 2 fits on the signal strength dataset. These bounds can be quantified by excluding parameter space in the allowed couplings of the scalar using a Gaussian probability density function approach. We develop and apply such an approach in this paper.
The outline of this paper is as follows. In Section II we discuss the EFT framework we employ, and the implicit assumptions about the UV origin of the scalar field that are adopted when fitting the data with various free parameters. In Section III we review and discuss the manner in which we treat the scalar signal strength data and electroweak precision data (EWPD), while in Section IV we present results, based on our fit method, of the status of the SM Higgs hypothesis. In Section IV B we discuss some of the implications of Higgs signal strength parameters for beyond the SM (BSM) physics, expressed through model-independent free parameters. In Section V we discuss novel and complementary methods to identify the allowed parameter space, and in Section VI we conclude.

II. THEORETICAL FRAMEWORK
An effective chiral EW Lagrangian with a nonlinear realization of the SU(2) L × U(1) Y symmetry gives a minimal description of the (non-scalar) degrees of freedom of the SM consistent with the assumptions of SM-like SU(2) c violation and MFV. The Goldstone bosons eaten by the W ± , Z are denoted by π a (where a = 1, 2, 3), and are grouped as with σ a the Pauli matrices and v = 246 GeV. In this approach, the EW scale v, which sets the mass of fermions and gauge bosons is introduced directly into the Lagrangian. The Σ(x) field transforms Adding a scalar field h to this theory is trivial. One chooses h to transform as a singlet under SU(2) c and a derivative expansion of such a theory is given by [24][25][26]] with Although we use the notation h, we do not assume in principle that this scalar field is the Higgs, or that the scale v is somehow associated with the vacuum expectation value of this field -as this is what we seek to establish from the data. As is well known, the a and c j parameters control the couplings of h to gauge bosons and fermions, respectively, and therefore, play a crucial role in the phenomenology of single h production. Note that in previous fits, and in the majority of this work, the assumption c j y u,d ij ≡ c y u,d ij is used and no distinction is made between the rescaling of the h-coupling to the u i and d i quarks. We will relax this assumption later on. This Lagrangian is common in the study of composite models and its relevance has been emphasized recently in Refs. [24][25][26]. It is also appropriate to study a pseudo-Goldstone boson (PGB) emerging out of an approximately conformal sector [27,28], or as the low-energy EFT arising in many scenarios where the Higgs is a composite PGB that emerges from the breaking of a larger chiral symmetry group [29][30][31][32][33][34]. We emphasize that this EFT setup can be matched to many UV frameworks and, being quite general, we do not confine ourselves to any particular UV scenario. 2 We approach the data in this way to be as model-independent as possible. However, even specifying the free parameters that one will use to fit the data introduces implicit model dependence. One can nevertheless broadly characterize certain parameter choices. As this is an EFT, L ef f is non-renormalizable. Here (and throughout this paper) we take the cut-off scale to be Λ ∼ 4 πv/ |1 − a 2 |. Since we are concerned with the phenomenology of single scalar production, the higher-order derivative operators are suppressed by powers of O(m 2 h /Λ 2 ). As such, we are justified in neglecting such sub-leading effects in this paper. Non-derivative higherdimensional operators will also exist in general. When the h field is not assumed to have a UV origin such as the SM Higgs, and is simply considered to be a singlet field [that need not necessarily transform under the nonlinearly realized SU(2) L × U(1) symmetry], the leading operators in the expansion in inverse powers of Λ appear at dimension five and are given by Here g 1 , g 2 , g 3 are the weak hypercharge, SU(2) L and SU(3) c gauge couplings, respectively, and the different tensor fields are the corresponding field strengths with their associated Wilson coefficients c i . The scale Λ corresponds to the mass scale of the lightest new state that is integrated out, which we assume is proximate to Λ. We will neglect operators originating from CP-violating sources due to the lack of any clear evidence of beyond the SM CP violation in lower energy precision tests. Note that the operators in L 5 HD can be further suppressed compared to the effects of a, c on (single) scalar production when the scalar field has specific UV origins. This is the case for example when h is a PGB, see Ref. [24] for a detailed discussion.
When one assumes that h is embedded into an SU(2) L doublet -H -as in the SM, the operators in L 5 HD first appear at dimension six, and the coefficients are suppressed by an extra factor of v/Λ when considering single scalar production. In this case, the dimension six operator basis is also extended by the operator For phenomenological purposes it is convenient to rotate to a basis for the operators given by where F µ ν is the electromagnetic field strength tensor and c γ = c W + c B in the case of an SU(2) L singlet field, and In this manner, one can understand that the choices to retain the effects of higher dimensional operators (or not) in performing global fits introduces implicit UV dependence. In introducing higher dimensional operators, one is also explicitly assuming the existence of new states charged under at least a subgroup of the SM group. Alternatively, if NP is uncharged under the SM group but couples to the h 2 operator 3 , then it can impact h-phenomenology by inducing an invisible hwidth (when the scalar field takes a vacuum expectation value). This leads to the modification of the SM branching ratios for each decay into visible SM final states f via We will update our recent fit of Br inv to Higgs search data [41] in a later section, and use this fit as a diagnostic tool to test aspects of our fit procedure.
In general, the coefficients a, c, c γ , c g , Br inv · · · are arbitrary parameters subject to experimental constraints. The generic cases we consider are: • Composite/Pseudo-Goldstone Higgs/Dilaton scalar theories.
In this case, a, c are free parameters in general, although they can be fixed in particular UV completions. If the scalar field is a Pseudo-Goldstone boson, it is also appropriate to neglect higher dimensional operators. We will fit to subsets of the parameters {a, c, BR inv , c γ , c g } in what follows. As this is a more general framework than the SM Higgs, we will use this EFT in assessing to what degree the SM Higgs hypothesis is consistent with the data or if deviations into this parameter space can give a substantially better fit.
• The SM Higgs as a low-energy EFT.
When the low-energy EFT is just the SM, the field h becomes part of a linear multiplet reducing Eq. (2) to the SM Higgs Lagrangian. In this case a = b = c = d 3 = d 4 = 1 and b 3 = c 2 = 0, and the only effect of NP is through non-renormalizable higher-order operators.
The naturalness problem of the SM Higgs mass operator, and recent experimental hints of deviations in the observed properties of the (assumed) Higgs, motivates moderately heavy NP and the introduction of BSM parameters (c γ , c g , BR inv ) as free parameters. We study the constraints on these parameters in detail in this paper.
We will not attempt to relate the constraints obtained on the various parameters to any particular underlying model, other than the SM, in this paper. This choice is motivated by the current lack of other clear experimental evidence of BSM states to guide coherent model-building. The classes of models discussed above can be considered as motivating examples.

A. Signal-Strength Data
In this section we describe our method for globally fitting to the parameters discussed above, and incorporating the recently released 8 TeV data [1], updated 7 TeV results from ATLAS [42], the released 7 TeV CMS data [43], and the recently reported Tevatron Higgs search results [44].
This work builds on our previous fits [2,41]. We only summarize the main details of the fit procedure and method here. Many subsidiary details of the fit procedure can be found in these reference works.
We fit to the available Higgs signal strength data, for the production of a Higgs that decays into the visible channels i = 1 · · · N ch , where N ch denotes the number of channels. The label j in the cross section, σ j→h , is due to the fact that some final states are defined to only be summed over a subset of Higgs production processes j. The reported best fit value of a signal strength we denote byμ i 4 .
The global χ 2 we construct is defined via The covariance matrix has been taken to be diagonal with the square of the 1 σ theory and experimental errors added in quadrature for each observable, giving the error σ i in the equation above. Correlation coefficients are neglected as they are not supplied by the experimental collaborations. For the experimental errors we use ± symmetric 1σ errors on the reportedμ i . For theory predictions of the σ j→h and related errors, we use the numbers given on the webpage of the LHC Higgs Cross Section Working Group [45]. 5 The minimum (χ 2 min ) is determined, and the 68.2% (1 σ), 95% (2 σ), 99% (3 σ) best fit regions are plotted as χ 2 = χ 2 min + ∆χ 2 , with the appropriate cumulative distribution function (CDF) defining ∆χ 2 .
We assume, as in Ref. [2,13], that the signal strength µ i in a given channel i follows a Gaussian distribution with the probability density function (pdf) given by with one-sigma error σ i , and best fit valueμ i . This is the case as long as the number of events is large, > ∼ O(10) events, [13]. We normalize these pdf's to 1 in the interval (0, ∞).
In the framework of the SM, the predicted values of the µ i are the same (equal to 1), and a universal signal strength modifier µ can be defined and applied to all channels. By multiplying together the individual channel pdf's (or the pdf's of a single channel reported at two operating energies), we can also define a combined PDF for µ. This can be done for each separate experiment or for a global combination of all experiments. The combined PDF is also Gaussian and has combinedμ c and σ c values given approximately 6 by We will use these relations to reconstruct the unreported 8 TeV data from the reported 7 and 7 + 8 TeV data. 7 Armed with combined PDF(s), we can determine the 95% C.L. exclusion upper limits on the signal strength parameter µ (µ < µ upL ) [2,13]. (We shall explain momentarily how to introduce an overall signal strength parameter in models with non-universal theory-predicted µ i . In the discussion that follows we are implicitly assuming that we are considering setting limits on combined signal strength parameters although we frame the discussion in terms of µ. A similar analysis can be carried through channel-by-channel instead of on the combined channels.) Such limits can be set on the combined signal strength parameter µ c or for an individual channel's signal strength.
Withμ i 's settling around unity with σ i errors getting smaller and smaller due to increasing integrated luminosities, we can already start to set also lower limits on µ. The condition for such lower bounds, say at 95% C.L., to be meaningful is that the symmetric intervalμ ± δ 95μ containing 95% of the integrated probability hasμ − δ 95μ > 0, in which case µ dwL ≡μ − δ 95μ corresponds to the lower 95% C.L. bound on µ. In this same case, we will takeμ + δ 95μ > 0 as the upper limit µ upL .
The conditions that define these limit are therefore and where erf(z) is the error function. However, whenμ − δ 95μ < 0, we shift the 95% C.L. interval to the asymmetric one (0, µ upL ) and revert to the upper limit definition [2,13] 8 Next consider models that depart from the SM, like SM(a, c) with couplings of the Higgs to fermions and gauge bosons modified by the a, c factors as explained above. The predicted values of the µ i 's will deviate from 1 in a channel-dependent way. In order to keep the same expected value of the signal-strengths for all channels, it is convenient to normalize the µ i 's in Eq. (9)  To summarize, we can exclude (at 95 % C.L.) a given scenario not only if it predicts too many signal events which are not seen (µ > µ upL ), but also if it predicts too few events, incompatible with the observed excesses associated with the reported discovery. The significance of such lower bounds will grow with more luminosity. Significances above 5σ become possible (by definition) after discovery (which excludes the background hypothesis µ = 0). We will plot both of these bounds mapping the allowed µ i upL , µ i dwL into the relevant parameter space through the dependence of µ i on the free parameters in our fit results. Figure 6, in Section V, shows examples of such lower limits.

B. Electroweak Precision Data
We incorporate EWPD [47][48][49] by adding it directly to the χ 2 measure in Eq. (10). When a is considered a free parameter, the shifts of the oblique parameters S and T are given by [32] ∆S The numerical coefficient is determined from the logarithmic large-m h dependence of S, T given in Ref. [48]. 9 As for EWPD, recent updates to the measurement of m W at the Tevatron [50,51] have refined the world average [52], and have significantly reduced the quoted error. Incorporating 10 these new measurements we use [53] S = 0.00 ± 0.10, while the matrix of correlation coefficients is given by Here we have assumed m h = 125 GeV, as corrections for shifting these results by a few GeV are negligible. There is a strong preference for a 1 in the global fit when EWPD is used, and the constraints on the scalar field can be directly associated with EWPD bounds. Note that the slight preference for a > 1 in the best fit region when EWPD is taken into account is subject to uncertainties in cut-off scale effects. Although the shift in the best fit point is of interest as a probe of possible new physics, we cannot clearly disentangle such a hint from cut-off scale effects.

A. Status of the Higgs hypothesis
The excess of events of ≈ 5σ significance reported by ATLAS and CMS peaks, as a function of the scalar mass, at slightly different values: 126.5 GeV for ATLAS and 125 GeV for CMS. 9 Here we have introduced an Euclidean momentum cut-off scale Λ. The degree to which Λ properly captures the UV regularization of S and T is model-dependent. We assume that directly treating this cut-off scale as a proxy for a heavy mass scale integrated out is a good approximation, i.e. that further arbitrary parameters rescaling the cut-off scale terms need not be introduced. 10 We thank J. Erler for kindly providing these EWPD results.
This difference can be attributed at this stage of the search to statistical fluctuations in the data.
Monte-Carlo studies [54] indicate that such effects can shift the observed maximum signal strength compared to the true signal strength maximum by ∼ 2 − 3 GeV. Due to this, a fit that combines data from different experiments at the same mass value might be biased and not necessarily better than using data at slightly different masses. As much more detailed data is available to us at the mass peaks, in our global fits we will use all available µ i taken at m h = 125 GeV for the CMS and the Tevatron (which has a ≈ 3σ excess over a wider region of masses), and at m h = 126.5 GeV for ATLAS (see Fig. 7 in the Appendix). 11 In order to assess the degree of consistency of the data with the SM hypothesis we first consider the effective Lagrangian given by L ef f and assume that higher dimensional operators are sufficiently suppressed so that c γ , c g can be neglected. We then perform a two-parameter χ 2 (a, c) fit and examine the ∆ χ 2 for the SM point (a, c) = (1, 1) compared with the best fit point. This defines a C.L. corresponding to the deviation of the SM hypothesis compared to the best fit point.
The result is shown in Figure 1 (left). We also show in Fig. 1 (right) the best fit regions when EWPD is added to the global χ 2 measure. Notice the dramatic reduction in the size of the best fit region along the a-parameter, which is forced to lie close to 1. These results visually summarize the current experimental status of establishing the Higgs hypothesis.
When EWPD is not used, the SM Higgs hypothesis of (a, c) = (1, 1) is < ∼ 2 σ (C.L. of 0.88) away from the best fit point, which sits at (a, c) = (1.1, 0.68). Note that here and in the following discussion we are choosing to round the C.L. and the best fit points. This is due to the preliminary nature of the 7 + 8 TeV data and is not limited directly to this accuracy due to the fit procedure we have adopted. The C.L. of the SM hypothesis in the combined data is consistent with our past results at 7 TeV [2]. With the updated data released and incorporated in our fit since version one of this paper the parameter space that has the global minimum has changed from c < 0 (with initial ICHEP data) to c > 0 (with post-ICHEP updates). 12 The existence of the non negligible parameter space with c < 0 is easy to understand. Due to the interference term in the h → γ γ decay width which is ∝ −ac, a negative c allows a relatively larger excess in γ γ events due to constructive 11 Recent updates from the experimental collaborations that have appeared since version one of this paper have released more information at m h = 125.5 GeV for CMS [55] and at m h = 126 GeV for ATLAS [56]. However, as the amount of information relevant to the fits we will perform released to date is still greater at the mass scales m h = 125 GeV for CMS and m h = 126.5 GeV for ATLAS we retain these mass choices in our fit. 12 The lack of data released for these subcategories to date at m h = 126 GeV is the primary reason we retain the use of m h = 126.5 GeV for ATLAS data. which restrict the fit to the region c > 0 (physically different from the region c < 0) 13 . As the absolute minimum of the χ 2 lies in fact in the discarded region, the shape and size of the 68% and 95% C.L. regions presented by CMS differ from the ones that we obtain in figure 2. If we also restrict our parameter space to positive c we have checked that we get excellent agreement with the CMS result. Note however that there is no valid reason to discard a priori the negative c region which offers in fact the interesting possibility of giving a good fit to the data by an enhancement of the hγγ coupling through constructive interference of the top and W loops.

Implications for an invisible width
The results of the last section can be interpreted as (partial) evidence in support of the SM Higgs hypothesis. Assuming then that the observed boson is the SM Higgs, we can study possible 13 We can always set a > 0 by a redefinition of the h field. The sign of c could in principle be changed also by rotating the fermion fields, but this would affect in the same way the fermion mass so that the sign of c/m f is in fact fixed and physically meaningful.  Table I  Here we quote ± symmetric 1σ errors.
The left plot of Fig.3 shows the result of extracting Br inv using our fit approach using two different methods. This allows an important cross check of our procedure and results. We have employed in our fit two approximations that require further justification. Using the assumption of Gaussian PDF's, we have extracted the 8 TeV data from the known 7 TeV data and the released The main difference between the results presented in Fig.3 with respect to the first version of this paper comes from the use of the ATLAS combined signal strength reported in Ref. [56], that

Implications for c g , c γ
One can also infer the current experimental bounds on the BSM parameters c g , c γ . We expect that these operators arise at the loop level, so we rescale the Wilson coefficients as c j =c j /(16π 2 ) for j = g, γ. Using the results of Ref. [59], the effects of these operators are incorporated as rescaling factors used in the fit and given by Here we have used m t = 172.5 GeV, m b = 4.75 GeV, m h = 125 GeV and α s (172.5 GeV) = 0.107995. When the Higgs has SM renormalizable couplings, then a = c t = c b = 1. We have retained the two-loop QCD correction to the SM matching of the h G A µ ν G A µ ν operator in the m t → ∞ limit in these numerical coefficients. The operators in L 5 HD also affect Br(h → γ Z), but the effects in our fit can be neglected. See Ref. [41] for further details and discussion on our fitting procedure to these higher dimensional operators.
We show in Figure 4 the results of fitting to c g , c γ using the current dataset when one assumes the SM values a = c = 1 (left); when one fits to c g , c γ , a, c and subsequently marginalizes over a, c (middle); and finally, when one fits to c g , c γ , BR inv and marginalizes over BR inv . These results summarize the preference in the current dataset for including higher dimensional operators when the scalar is not assumed to be the Higgs.
We also show in Fig. 5 (left) the fit to c g , c γ , a, c, marginalizing over the higher dimensional operators. This case shows the preference for a, c even when more massive states are integrated out of a composite scalar theory (for example) contributing to c g , c γ . We see that the SM hypothesis is significantly improved in its consistency with the dataset in the context of NP of this form. This is particularly relevant as it directly shows what the data tells about a, the key parameter that probes the involvement of h in electroweak symmetry breaking. A final plot in Fig. 5 (right) shows the allowed space when c t is varied independently from the remaining fermion couplings which are treated with a universal rescaling.
The fit to higher dimensional operators shows a preference for a BSM contribution to c γ , which deserves some comment. First, one can study the model independence of this preference for an enhancement of c γ by constructing a one dimensional χ 2 distribution, marginalizing over the unknown operator c g when the SM values a = c = 1 are assumed, or over (c g , a, c) when one is considering non-SM scalar scenarios. Performing this exercise we find that a preference for c γ = 0 only exists in certain cases where implicit UV assumptions are adopted due to the parameters used to fit the data. One can make a number of observations regarding enhanced h → γγ event rates.
The coupling to F µ ν can come about due to c W B . If this is the case, one can bound the allowed enhancement of µ γ γ due to related EWPD constraints. Using the relation [59][60][61] one finds the current experimental constraint |v 2c W B /Λ 2 | < ∼ N 0.63 for an N σ deviation considered in the EWPD parameter S, assuming m h = 125 GeV. Using µ γ γ ≈ R g R γ , this leads to the bound µ γ γ < 1.2 (1.4) for a one (two) sigma deviation in the S parameter when c g = 0. Restricting the unknown higher dimensional operators to have a global χ 2 measure in the 2σ allowed region (∆χ 2 (c g , c γ ) < 6.18), one can maximize the enhancement of µ γγ considering the related constraints on c W B . One finds that an enhancement of 1.4 is possible when a one sigma deviation in the S parameter is also allowed. The c γ Wilson coefficient could also come about due to c B , c W or a combination of the two. In this case, the EWPD constraint does not directly apply, and an enhancement of µ γ γ by a factor of < ∼ 1.7 is still allowed when maximizing the contribution subject to the constraint ∆χ 2 (c g , c γ ) < 2.3.
This analysis has assumed no relationship between the Wilson coefficients c γ , c g , which is fixed in any particular UV completion. It is interesting to note that in general when integrating out a single BSM field one expects a strong relationship between the Wilson coefficients, with identical loop functions in many cases, the only differences in the matching onto the Wilson coefficients is due to the relative charges of the new states under the SM subgroups SU(3) c and U(1) em .

V. STUDIES OF CONSISTENCY AND TENSION WITHIN THE DATASET
The consistency between the search results (μ i ± σ i ) from different channels can be quantified as follows. For each channel i we construct its Gaussian approximation to the pdf of the signal strength, pdf i (µ) = pdf(µ,μ i , σ i ), which we can contrast with the full combined PDF (either in a given experiment or for the overall combination of all data), PDF(µ) = pdf (µ,μ c , σ c ). For each single channel we calculate its p-value with respect to the global combined PDF, i.e., p ic is the p-value forμ i assuming the full PDF(µ). We can also define the p-value of the globalμ c with respect to the individual pdf i (µ), which we will denote by p ci . In order to treat properly the pvalues for cases withμ i < 0, we will normalize the pdf's in this section so that they give 1 when integrated over the whole interval µ ∈ (−∞, ∞). We then have, forμ i <μ c , and, forμ i >μ c , where erfc is the complementary error function, erfc(z) = 1 − erf(z). We will say channel i is in tension with the rest of the data if p ic and p ci are both very small. For a given critical p-value p N , corresponding to N standard deviations, channel i is not consistent with the combined dataset at (1 − p N ) C.L. if p ic , p ci < p N and the model can be excluded based on that disagreement. The consistency condition reduces simply to We will choose N = 2 in our discussion.
When this test for consistency is applied to the (μ i ± σ i ) dataset, interpreted as coming from a SM Higgs signal, we find tension at this 2σ level for two ATLAS γγ channels at 7 TeV, those labelled URhPTh and CChPTt (the two outliers easily identified in Fig. 7). In extensions of the SM, like the two-parameter scenario SM(a, c) we have discussed in previous sections, the rescaling of the different channels (which affectsμ i ± σ i as explained in section III A) can introduce very significant distortions in the pdfs and cause too large tensions for some other channels. Such regions of parameter space can therefore be excluded on this basis. Figure 6 illustrates this by showing, on addition to the best fit regions and the 95% C.L. exclusion regions derived from upper and lower limits on the signal strength, the regions in (a, c) space that would be excluded due to more than 2-σ tension in some search channel (purple shaded regions delimited by straight lines).
We show such limits both experiment by experiment and for the combined result. Typically, in the excluded regions several channels at the same time cause the exclusion. We show in each case all the region that is excluded by at least one channel (with the exception of the ATLAS case, see below). In each case, the channels that have bigger exclusion power are the following. For Tevatron, the bb channel; for CMS, at 7 TeV, γγ jj and at 8 TeV, γγ 3 and τ τ ; for ATLAS, the two γγ channels mentioned above, URhPTh and CChPTt at 7 TeV, cause 2σ tension in all the region of parameter space shown and we do not mark this area. Besides these channels, there is 2 σ tension also from γγ CClP T t at 7 TeV and from W W → llνν at 8 TeV. The fact that the tension limits are straight lines passing through the origin is due to the fact that any common rescaling of a and c (that leaves c/a invariant) also leaves invariant the tension associated to any channel as the latter are functions of ratios (µ i − µ c )/σ i,c , and such ratios are also functions of c/a. We see from Figure 6, that fermiophobic scenarios, along the axis c = 0, are excluded at this level of confidence.
This channel-tension analysis is clearly related to the χ 2 study we have also performed and the tension exclusion limits tend to exclude regions of parameter space that give a bad fit to the data. However, this approach seems to be more powerful in being able to exclude definitely some models. We also see that there are regions of parameter space that are not excluded by the conventional upper (or lower) limits imposed on the signal strength parameter, yet can be excluded by the tension exclusion limit. This is simply due to the fact that the dataset can contain two widely separate channels and still give a combined PDF that respects the upper limit, which is not able to probe in such cases the internal inconsistency of the individual channels. We conclude that this type of analysis offers a complementary tool in testing model performance.

VI. CONCLUSIONS
We have studied the evidence of a scalar field that has been discovered by the LHC collaborations using an effective field theory framework. For those cases in which only the combined 7 + 8 TeV signal strengths are available (currently CMS combinedμ τ τ ,μ W W used in the fit) in addition to the 7 TeV results, we make use of Eq. (12) to reconstruct the unreported 8 TeV data. Note that this relies in the use of Gaussian approximations to the PDF's to describe the data (and signal strengths) and should be increasingly accurate as the total number of events increases. This can be done without knowing directly the experimental likelihood function in the limit that correlations are neglected, which is an assumption we are already forced to adopt as this information is not supplied by the experimental collaborations. We show in Fig. 7 the resulting reconstructed data.
An interesting check of our approach is to use a subset of the provided subclass signal strengths to reproduce a reported combined signal strength. This exercise can be carried out, for example, on the supplied vector boson fusion tagging bb signal strength and tt h signal strength (that uses h → bb) and comparing to the presented combined h → bb signal strength. We have carried out this procedure for the presented CMS data and find good agreement with the reported results, within our estimate of 5 − 10% error introduced due to a lack of correlations.
We have updated our fit from v1 of this paper to include the following information that has been released since the first version appeared on the archive. The full subclasses of γ γ events are now available from ATLAS and CMS at 7, 8 TeV. Also, the production channel composition of the γ γ subclasses have been supplied [63,64]. We have modified our fit to use this information consistently with our rescaling procedure. This procedure replaces our utilization of estimated gg contamination in the pp → γ γ jj signal events. Also note that despite the cuts of the "tight" and "loose" dijet channels of CMS indicating they are not mutually exclusive event classes, the CMS collaboration vetoes an event appearing in the "loose" pp → γ γ jj sample if it passes the "tight" cuts [65]. As such, these event classes can both be included in the global fit we perform. 15 Finally we note that the 7, 8 TeV signal strengths for the bb and ZZ signal strengths have now been supplied and are incorporated in our fit. Comparing our extracted ZZ 8 TeV result to the experi- 15 We however still thank V. Sanz for kindly providing the contamination coefficients for v1 appropriate for our past procedure.The contamination in the first version of this paper was reported with a typographical error, the correct contaminations are = 0.032 for the 7 TeV dijet tagged diphoton signal, = 0.023 for the 8 TeV "tight" dijet tagged diphoton data and = 0.039 for the 8 TeV "loose" dijet tagged diphoton data [62]. Here is the contamination of the pp → γ γ jj signals due to gg Higgs production events, when is defined such that the rate is given by ( σ gg→h + σ jjh ) × Br(h → γ γ).
mentally supplied number we find agreement within the estimated accuracy of our procedure. For the bb CMS signal strength we note that the 7 TeV signal strength recently reported in Ref. [66] differs from the previously public 7 TeV signal strength.  Red: reported 8 TeV data, or reconstructed 8 TeV data.