Probing for Invisible Higgs Decays with Global Fits

We demonstrate by performing a global fit on Higgs signal strength data that large invisible branching ratios Br_{inv} for a Standard Model (SM) Higgs particle are currently consistent with the experimental hints of a scalar resonance at the mass scale m_h ~ 124 GeV. For this mass scale, we find Br_{inv}<0.64 (95 % CL) from a global fit to individual channel signal strengths supplied by ATLAS, CMS and the Tevatron collaborations. Novel tests that can be used to improve the prospects of experimentally discovering the existence of a Br_{inv} with future data are proposed. These tests are based on the combination of all visible channel Higgs signal strengths, and allow us to examine the required reduction in experimental and theoretical errors in this data that would allow a more significantly bounded invisible branching ratio to be experimentally supported. We examine in some detail how our conclusions and method are affected when a scalar resonance at this mass scale has couplings deviating from the SM ones.


I. INTRODUCTION
Two outstanding questions of importance that the LHC should shed light on are the origin of electroweak symmetry breaking (EWSB), and the relationship of the mechanism of EWSB to new states beyond the Standard Model (SM).
There is strong indirect evidence for the EWSB sector being described by a theory that includes a particle that (at least) approximately has the properties of the SM Higgs boson. This evidence follows from many observables in flavour physics, from electroweak precision data (EWPD), LEP, the Tevatron and now the LHC. The SM Higgs is consistent with the results of these experimental probes in its pattern of breaking custodial symmetry (SU(2) c ) [1][2][3] as well as in the manner by which it sources the experimentally established pattern of flavour violation. In light of these arXiv:1205.6790v2 [hep-ph] 5 Nov 2012 results, there is substantial indirect evidence that a scalar field involved in EWSB will also be SM Higgs like in that the soft Higgs theorems of Refs. [4,5] will be approximately respected, i.e. the scalar field will couple to the SM fields with a strength that is proportional to the mass of the corresponding SM particle. the Higgs hypothesis is not yet established. In particular, there remains a significant freedom in the allowed couplings of a scalar effective field to the SM gauge bosons and fermions -so long as such a resonance has the approximate symmetries and properties discussed above [8][9][10]. 1 Such deviations in the properties of a scalar field from the SM Higgs can be interpreted as following from the Higgs boson emerging from a strongly interacting sector as a pseudo-Goldstone boson, or as the leading effect in the effective theory of more massive states that are integrated out.
In light of this experimental situation, attempting to use current (and future) Higgs signal strength parameters to establish relationships between the EWSB sector and beyond the SM states is speculative. This is certainly the current status, as the suggestive clustering of signal excesses at m h ∼ 124 GeV has not risen to the level of experimental evidence for a scalar resonance.
Nevertheless, in this paper we will assume that future data will support the discovery of a scalar resonance at approximately this mass scale. Further, we will consider current signal strength measurements as indicative of the properties that such a scalar resonance has when performing global fits. In anticipation of such a discovery, it is of interest to consider how to efficiently extract evidence of yet other states coupled to such a scalar field.
The gauge invariant mass operator of the scalar degrees of freedom, being of dimension two, is expected to couple generally to all degrees of freedom. It is difficult to forbid a coupling of this operator at the renormalizable (or non-renormalizable) level to new states. As such, the measurement of the decay width of a new scalar resonance to states that do not directly lead to significant excesses in the Higgs discovery channels, defined in this paper as its invisible branching ratio Br inv , could be the first direct measurement of interactions with states beyond the SM. This exciting possibility has lead to many studies on extracting the invisible width of the Higgs. In this paper we will explore a very straightforward route using current and future experimental results on Higgs properties, expressed in terms of signal strength data, to probe for evidence of a Br inv . We first show (in Section II) that the current experimental hints of a new scalar resonance at m h ∼ 124 GeV do not put strong constraints on Br inv . We then explore in detail how to extract evidence for (or exclude) a Br inv using global combinations of best fit signal strength parameters, performing global χ 2 fits, and demonstrating a global probability density function (PDF) approach that can be used to explore and optimize searches for Br inv in certain scenarios of beyond the SM (BSM) physics (Section II A). In Section II B, we examine the related issue of the precision with which Br inv is expected to be known in these scenarios when errors are small enough for a resonance discovery to be claimed.
These promising results raise the question of the robustness of such a global approach. These techniques are most promising when BSM physics couples to the SM primarily through the 'Higgs portal' [14][15][16][17][18][19], i.e. they are optimal in BSM scenarios where new states are not charged under the SM gauge group and couple to the SM (initially) through the SM gauge singlet scalar mass operator. This is the case we explore in detail throughout Section II. We briefly discuss and summarize the prospects for global fits to uncover Br inv in broader scenarios in Section III where the effective couplings of the Higgs to the SM fields deviate from their SM values due to the Higgs being a pseudo-Goldstone boson or due to the presence of higher dimensional operators. We present detailed numerics for these scenarios (based on current global Higgs fits to experimentally reported signal strengths) throughout the remainder of Section III.
We find that global fits to signal strength parameters will be a powerful approach to search for evidence of Br inv in the scenarios we consider. But we note that challenges will exist in disentangling other new physics effects. This will likely require a combination of indirect global fit approaches to extracting Br inv , which is the focus of this paper, and more traditional direct searches for Br inv based on kinematic properties of Higgs signal channels. We compare and contrast these approaches in Section IV and then discuss our overall conclusion in Section V.

II. THE STANDARD MODEL HIGGS AND Br inv
In this section, we will present the current global best fit 2 results for Br inv = Γ inv /(Γ inv + Γ SM ) for the SM Higgs, where Γ inv is the decay width to 'invisible' states, as defined above, and Γ SM is the decay width of the SM Higgs. We perform a global fit to the available Higgs signal data, fitting to fifteen Higgs signal strength parameters µ i reported by ATLAS, CMS and the Tevatron collaborations which are defined as for a production of a Higgs that decays into the visible channel i. We use the best fit values of µ i , denoted byμ i , as reported by the experimental collaborations. The label j in the cross section, σ j→h , is to denote that signal events in some final states are defined (by selection cuts) to only be summed over a subset of Higgs production processes j. We construct a global χ 2 measure on thê µ i by defining the matrix C as the covariance matrix of the observables, and ∆ θ i = µ i −μ i as a vector of the difference between the signal strength variable µ i and the best fit value of the signal strengths 3 , Here i = 1 · · · N ch , where N ch denotes the number of channels. The matrix C is 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 (currently not supplied by the experimental collaborations) are neglected. For the experimental errors we use ± symmetric 1 σ 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 [20].
The minimum (χ 2 min ) is determined, and the 68.2% (1 σ), 95% (2 σ) best fit confidence level (CL) regions are given by ∆χ 2 < 1, 4, respectively, for χ 2 = χ 2 min + ∆χ 2 . Here, the CL regions are defined by the cumulative distribution function (CDF) for a one-parameter fit. 2 See Ref. [10] for a detailed discussion on our fitting procedure. The values of the fifteen inputμ i used are reported in the Appendix for completeness. Note that throughout this paper we will assume a 3% contamination due to gg events in the γγj j signal strength, see the Appendix for further details. 3 In a simple counting experiment the definition isμ = (n obs − n b )/n SM s , in terms of the observed number of events (n obs ), the number of background events (n b ) and the expected number of SM signal events (n SM s ).
Although we are fitting for evidence of new 'invisible' states, we do not include effects due to new unknown interactions on the production of the SM Higgs in this section 4 . We include an invisible width by modifying the SM branching ratios universally for each decay into final states f via Thus, the effect of including an invisible width (of BSM origin) on the signal strengths is that the expected µ i = 1 in the SM is modified to an expectation of µ i = 1 − Br inv . We fit for the parameter Br inv assuming a SM Higgs with a total SM width Γ SM and a particular Higgs mass. The resulting χ 2 as a function of Br inv is shown in Fig. 1. Interestingly, we find that the global χ 2 is minimized for a non-zero value of Br inv . 5  It is instructive to look more closely at the χ 2 fit to all channels we have performed rewriting Eq. 2 as where we have introduced the combined variables Note that Eq. 4 is valid if all the µ i are equal, as is the case for the SM with an addition of Br inv .
This decomposition illuminates what our analysis of the fit to individual channels really does.
The location of the minimum of the fit, and the N σ intervals, is controlled by the first term in Eq. 4, which depends only on the combined parametersμ c and σ c but not on the dispersion of the differentμ i 's around their averageμ c . How good the fit is, is just given by the second piece in Eq. 4, which is simply χ 2 min , and does depend on how separate are the individual channelμ i 's fromμ c . Interpreting, as we do in this paper, deviations ofμ from its SM value of 1 as a Higgs invisible width, one immediately obtains that the χ 2 is minimized (defining Br min inv ) when This also offers the alternative approach of bypassing the individual channel analysis and using directly theμ c values reported by the experiments in Table I. We can use this data to do the χ 2 fit as the effect of Br inv on the signal strengths is a common multiplicative correction.   A. Global PDF approach to discovering Br inv in the SM The χ 2 approach we have discussed can be justified on the basis of a more detailed analysis that makes use of the combination of the PDF's for all sensitive Higgs search channels. To the extent that these PDF's are well described by Gaussian distributions, both approaches are basically equivalent in terms of the discovery reach afforded. The combined PDF approach however has a more direct physical interpretation and makes clearer the expected experimental sensitivity required to discover or bound Br inv . Moreover, this treatment is more powerful as it could also capture possible deviations from simple Gaussian shapes. On the other hand, the χ 2 fit is useful to determine if a reduction in Higgs signal yields in all channels is really universal and leads to a value of χ 2 min indicating a good fit, and is very convenient for taking into account the effect of imposing EWPD constraints. In this section, we will discuss global searches from the global PDF perspective, discussing the relationship between the errors inμ i and the discovery reach (or exclusion prospects) for Br inv with such an approach.
As in the previous χ 2 analysis, the key point to the power of global searches is the fact that, 6 in the presence of Br inv , the expected measured values of the strength of the signal with respect to pure SM expectations are modified as (µ i = 1) → (µ i = 1 − Br inv ). Due to this, one can construct a global PDF (combining the PDF's of individual channels) sensitive to this shift in the signal strengths which aids in experimentally distinguishing Br inv = 0 from the SM case Br inv = 0 . As in Ref. [9], we will assume that the PDF for each µ i reported by the experimental collaborations can be approximated by Gaussian distributions with one sigma error σ i , and best fit valueμ i for the signal channels i. This is the case (especially for µ nearμ i ) as long as the number of events is large, > O(10) events, [9] and systematic errors are subdominant. Using experimentally reportedμ i and σ i is the best and simplest way of approximating the likelihoods in the neighbourhood ofμ i . Experimental information on the 95% CL exclusion limits on µ i give additional information on the PDF's up to higher values of µ and allow one to identify channels for which there are non-Gaussian tails. In such channels, using exclusion limits to extract theμ i (as done in Ref. [21]) will tend to overestimate theμ i . This would subsequently bias an extracted value of Br inv based on such constructedμ i .
A global combination of all the visible channels PDF's, where every PDF is approximated as above, is obtained as a product of the pdf i (µ) (where i = 1 · · · N ch ) and it is also approximately where σ c andμ c are defined in Eq. 5 and Here we have normalized with the condition ∞ 0 pdf (µ,μ c , σ c )dµ = 1 and Erf is the standard error function. In the limit where σ i ≈ σ, and correlations are neglected, one has the simple To find evidence of a nonzero Br inv one has to be able to distinguish the Br inv > 0 hypothesis (dubbed SMinv) from the SM Higgs hypothesis with Br inv = 0, and discern this case from the background-only hypothesis. We use the same approach used routinely in experimental analyses to estimate the significance of a signal excess in the data, which quantifies how unlikely such an excess would be if interpreted as an upward fluctuation of the background. One defines a p-value for the background-only hypothesis as where the background probability density function (or likelihood) is approximately a Gaussian centered at µ = 0 (as n obs = n b ) with some globally combined 1-standard deviation spread (σ c ) that results from the combination of N ch different channels each with an individual σ i : 7 7 The overall normalization of pdf b (µ, σ) can be fixed by For the i th channel, with an expected number of background events n b,i and an expected number of signal events (in the SM) n SM s,i one has σ i = √ n b,i /n SM s,i (neglecting systematic effects). Again, in the limit where all σ i are comparable, and correlations are neglected, one has the simple scaling A p back -value as small as that corresponding to a 5σ fluctuation will be required to claim Higgs discovery 8 . See Fig. 2 for an illustration of the definition of the background p-value.
Besides having a small enough p back for a Higgs discovery, to claim evidence for Br inv > 0 we should be able to discard also the pure SM hypothesis (with Br inv = 0), as a downward fluctuation in the signal yield could be misinterpreted as Br inv > 0. We proceed exactly as before and construct the global PDF for the SM hypothesis as a Gaussian centred at the SM value µ = 1: Here N SM is implicitly defined by the condition Also, neglecting systematic effects, as in Eq. 11 so that we use the same notation for both. We then compute the p-value associated with the pure SM hypothesis as See Fig. 2 for an illustration.
Claiming evidence for Br inv > 0 requires having simultaneously a small p back and p SM . In order to quantify this, notice that experimental collaborations in Table I. Figure 3, left plot, shows the p-values for both hypothesis for a Br inv measurement with certain precision σ c , chosen for illustration at around its current value ∼ 0.3 and future values σ c = 0.15 and 0.05. As expected, claiming evidence for Br inv > 0 will be easier for Br inv ∼ 0.5 and will be facilitated by a reduction in σ c . With the current value, the plot also shows that the weak indication of Br inv ∼ 0.12 is perfectly compatible with a downward fluctuation of a SM-like Higgs, and even compatible with a background upward fluctuation at ∼ 3σ. Figure 3 (right), shows the required precision σ c for 1σ to 5σ evidence of nonzero invisible width. As expected, the ability to find evidence of a nonzero Br inv degrades for small values of this parameter, when it is harder to disentangle SMinv from the SM and also for Br inv → 1, when it is hard to discern a small signal over background. 9 As σ i = √ n b,i /n SM s,i , with more luminosity the statistical component of σ c will scale down with ∼ 1/ √ L (so that the plotted 1/σ 2 c increases linearly with L). As an example, assuming this scaling of σ c , and that the best fit value of Br inv = 0.12 obtained in the global fit is the true value, this indicates that accumulated signal events should be increased (compared to the current data set) by a factor of ≈ 25(100) to reach 2(4)σ evidence of this Br inv .
At the end of the current LHC run it is expected that the accumulated luminosity will be enough to reach the level required for a 5σ SM Higgs discovery per experiment. This means that both σ c,ATLAS and σ c,CMS will be down to ∼ 0.2 or lower. (This expectation is consistent with recent public statements by CMS and ATLAS, see Ref. [22].) Taking such values for these quantities, and combining with the current σ c,Tevatron , we arrive at σ c 0.15 (half the current value) as a reasonable number to expect by the end of the year.

B. Bounding Br inv in the SM
With the measured overallμ c , known with some error σ c , we can also set 95% CL limits on Br inv . For this purpose one can use the overall PDF (from the combination of all Higgs search channels) for the signal strength parameter, which we again approximate by a Gaussian 9 We have numerically cross checked the relationship between sensitivity to BR inv and σ c shown in the p-value results with another simple test based directly on the lack of overlap of global PDF's. Introducing a PDF for the background only scenario, a SM PDF, and a test theory PDF, simply insisting that the N sigma allowed µ in the test theory PDF lies outside of the N sigma allowed regions of the other two PDF's, one finds a similar sensitivity to what is indicated for a 2 N σ evidence for a common BR inv in the p-value test. centred atμ c with standard deviation σ c . As we will interpretμ c < 1 as coming from a nonzero Br inv we restrict now µ to the interval (0, 1) and normalize the combined PDF accordingly, i.e.
imposing the condition that the interval is centred atμ c if µ L1 > 0 and µ L2 < 1. Otherwise one fixes µ L1 = 0 or µ L2 = 1. From this interval we derive a 95% CL allowed band for Br inv as One can also place a comparable bound in the context of a χ 2 fit that is given by The sensitivity of this χ 2 -based bound is expected to be equivalent to the sensitivity to BR inv in the PDF test in the Gaussian limit. Figure 4 shows the sensitivity band, as a function ofμ c for the PDF test, for several values of its error σ c : the current one (σ c = 0.3); the combined error expected when both ATLAS and CMS accumulate enough data for a 5σ Higgs discovery per experiment over this year (σ c = 0.15), with the 95% CL excluded region shaded; and with a future error value down to σ c = 0.05. As expected, ifμ c is small this requires a large invisible width and a lower limit on Br inv can be set while, ifμ c is closer to 1, then only an upper limit on Br inv can be derived. Note that for Br inv → 0 or 1, the corresponding values ofμ c themselves require smaller σ c than for moderate values ofμ c to reach a discovery. Thus if a discovery is actually made in these cases, any corresponding Br inv will be simultaneously more accurately known than for theμ c ∼ 0.5 case.  This can change with higher energy/luminosity and the right plot shows a hypothetical future situation with nonzero Higgs invisible width after collection of more data (such that the current σ c ∼ 0.3 used in the left plot is reduced by a factor 5). Besides a hypothetical curve with the best value for Br inv , the plot also shows the regions of parameter space for which p SM and p back are below 5σ, illustrating how such an analysis could claim indirect evidence for Br inv = 0.

III. ROBUSTNESS OF GLOBAL FITS TO EXTRACT Br inv
In the previous section we have examined the prospects for bounding or discovering Br inv for the Higgs in BSM scenarios where new physics primarily couples to the dimension two scalar mass operator. In this section, we will examine how robust these conclusions are when a scalar resonance that has only approximately SM Higgs properties is involved in EWSB. First we will consider in Section III A the case of a minimal effective chiral EW Lagrangian with a non-linear realization of SU(2) L ×U(1) Y and a light scalar resonance. This scenario is most easily interpreted in composite Higgs scenarios and introduces parameters (a, c) for the unknown coupling of a scalar resonance to the gauge and fermion fields of the SM, with the SM case corresponding to (a = 1, c = 1). (See Refs. [25][26][27]). The obvious problem one faces in this case is how to determine if a universal reduction in signal yields is due to a nonzero Br inv or to a uniform reduction of the Higgs couplings involved in the search channels (i.e., a = c < 1).
The robustness of global fits for Br inv in the presence of unknown higher dimensional operators is also important to determine. In Section II, when considering the effects of Br inv on the SM with Here we have neglected potential terms that are not relevant for the fits we will perform. Fitting the current data in such a theory has been recently explored in the literature [9,10]. Note that the SM Higgs is a special case of this theory, and corresponds to a linear completion (h becomes part of a linear multiplet) of this non-linear sigma model, with a = c = 1.

B. Imposing EWPD
It is useful to consider fitting to EWPD simultaneously with the global data to obtain a more constrained parameter space in this effective theory. 11 In EWPD analyses, the corrections to the gauge boson propagators in this effective Lagrangian can be expressed in terms of shifts of the oblique parameters S and T [30][31][32] given by These equations are approximate in that the numerical coefficient is determined from the logarithmic large m h dependence of S, T given in Ref. [31]. Here we have introduced a Euclidean momentum cut-off scale Λ, which approximately represents the mass of new states that are required to cut-off the growth in the longitudinal gauge boson scattering. In a full calculation, with all degrees of freedom, the cut-off scale will cancel. The degree to which this Euclidean cut-off properly captures the UV regularization of these integrals by new states not included in the effective theory is model dependent. We assume that the UV completion of the effective Lagrangian is such that directly treating this cut-off scale as a proxy for a heavy mass scale integrated out is valid, i.e. that further arbitrary parameters rescaling the cut-off scale terms need not be introduced.
The cut-off scale is chosen to be Λ = 4 π v/| √ 1 − a 2 | for a = 1. 11 In fact, we will find in subsequent sections that marginalization over multiple parameters in the three dimensional fit space including EWPD is required to obtain a residual χ 2 distribution that is not flat.
For EWPD we use the results of the Gfitter collaboration [33] for m h = 120 GeV, S = 0.04 ± 0.10, T = 0.05 ± 0.11, U = 0.08 ± 0.11, and the correlation coefficient matrix is given by We shift these results to having the input m h = 124 GeV using the one-loop contribution of the SM Higgs field to S and T. This numerical shift is < ∼ 10 −2 . There is a strong preference for a 1 in a global fit due to EWPD, i.e the SM mechanism of mass generation of the W ± and Z is strongly preferred in minimal scenarios where EWPD can be directly interpreted to dictate the value of a. When EWPD is imposed one has a bias in the fit space so that a > 1, but this should not be over-interpreted. This bias could in principle be a hint for the existence of other states in EWPD, but this possibility cannot be disentangled from cut-off scale effects without further experimental and theoretical input. We conservatively consider this bias to be simply a numerical artifact of our cut-off procedure.
Interestingly, EWPD offers a handle to disentangling the degeneracy between a = c < 1 and the presence of Br inv : a = 1 has a direct impact on EWPD, while the new singlet states (into which the Higgs can decay invisibly) can have no impact on EWPD. In any case, the possibility of such degeneracy implies that further cross-checks of Br inv > 0 would be needed to confirm an eventual indirect evidence coming from the global tests discussed in the last Section. Directly confirming such indirect evidence for Br inv is best accomplished in more traditional studies of experimental sensitivity to Br inv based on the kinematics of Higgs decay products. We discuss prospects for such a direct confirmation in Section IV.

C. Marginalizing/Fixing Parameters
First, consider the case of fixing or marginalizing over one of the parameters (a, c, Br inv ) 12 to examine the robustness of our global fit results when Br inv = 0 is assumed, as in Ref. [10]. Fits 12 In the remainder of the paper we will always choose the value m h = 124 GeV as we have shown that the fit results are not strongly dependent (considering current errors) on the chosen mass (when varied in the range 124 − 126 GeV). plots EWPD is also imposed, while in the lower plots EWPD is not included in the global fit. Here the green region is the 65% CL region defined through the CDF for a two parameter fit. The yellow region is the 90% CL region and the grey region encloses the 99% CL region. Also shown as solid black lines are the 95% exclusion regions (outside this line is excluded at 95% CL) in the parameter space using the procedure described in Appendix B of Ref. [10] and the data in the Appendix.
with various Br inv as an input value are shown in Fig. 6. We find that, when EWPD constraints are incorporated into the fit, the c < 0 minimum of the χ 2 is preferred for larger values of Br inv . This is easy to understand, as Br(h → γγ) depends on the interference of fermion and gauge boson loops with an interference term ∝ −a c. As the invisible width gets larger and the expected number More precisely, the lower plots shown in Fig. 6 have a simple scaling property corresponding to a dilatation from the origin in (a, c) space, relating the spaces in plot i to plot j as The constraints from EWPD on the fit space can be more easily understood by directly comparing the fit spaces as shown in Fig.7 (left). Similarly, the dilatation scaling of the best fit space is illustrated in Fig.7 (right) where we plot the fit space as a function of the scaling variables Now consider treating each one of the parameters (a, c, Br inv ) as a nuisance parameter in turn.
Doing so we can also examine the effects of an unknown parameter on the remaining fit space. For example, in marginalizing over Br inv we define a reduced χ 2 function χ 2 (a, c) = χ 2 (a, c, Br inv (a, c)), where Br inv (a, c) is given by the solution of dχ 2 (a, c, Br inv )/dBr inv = 0. Then the allowed parameter space is defined through the CDF for a two parameter fit, and we obtain the results in Fig. 8. Marginalizing over the parameters c and a we find the results shown in Fig. 9 and Fig.   10 respectively which demonstrate the correlation between the allowed Br inv , and the allowed parameter space for the remaining unknown parameters. This correlation is due to the dilatation relationship shown in Eq. 23.
Finally one can marginalize two of the free parameters simultaneously in order to obtain the residual χ 2 distribution to examine if the slight statistical preference for Br inv > 0 persists. In this case, one must impose EWPD to avoid a flat distribution in the remaining free parameter. We find the results shown in Fig. 11. Of most interest is the result of marginalizing over free gauge and fermion couplings, while imposing EWPD. In this case, one finds that the global fit in this theory is now Br inv = 0, with the 95% CL limit Br inv < 0.57, for a scalar mass of 124 GeV.
Comparing this to the SMinv result of Section II, we see that an unknown a, c can remove the slight preference for Br inv = 0 in the current global fits when the χ 2 min with c > 0 is the global minimum. Conversely when the χ 2 min with c < 0 is chosen, the slight preference for Br inv in the  When the c < 0 minima is chosen, and the marginalization is performed, the preference for Br inv in the current data increases due to the interference effects previously discussed.
data set is not removed. Then the best fit is Br inv = 0.21, with the 95% CL limit Br inv < 0.75. Assuming that these BSM states do not source CP violation, the operators of interest for Higgs phenomenology (in global fits toμ i ) are given by Note that g 1 , g 2 , g 3 are the weak hypercharge, SU(2) gauge and SU(3) gauge couplings and we are using the notation of Ref. [34]. The scale Λ corresponds to the mass scale of the lightest new state that is integrated out. We are primarily interested in the effects on σ gg→h and Γ h→γγ as these are loop level processes in the SM, sensitive to BSM effects. As we expect loop level contributions to these operators from the BSM states, we rescale the Wilson coefficients as c j =c j /(16π 2 ) for j = G, W, B, W B and fit to combinations ofc j .
Using the results of Ref. [34], the effect of these operators are Normalizing in this manner is done to reduce the SM dependence in the BSM correction when this rescaling is used in our fits, and a numerical value is used for σ SM gg→h . As in Ref. [34], 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. Due to this choice, this correction cancels out (in the m t → ∞ limit) of the overall coefficient of the BSM effects when multiplied by the numerical value of σ SM gg→h . This is a ∼ 10% correction on the quoted numerical coefficient. Initial state radiation and vertex corrections to G A µ ν G A µ ν are expected to be common multiplicative factors for the operator h G A µ ν G A µ ν in the m t → ∞ limit, and as such are not incorporated in the numerical factors multiplyingc G ,c γ above. We will consider the parameter space where the SM is modified by these corrections and Br inv = 0 in this section, fitting to (v 2c relationship between these parameters, if any, is model dependent and unknown. As such, we fit to the data assuming no relationship between the three parameters. 13 The operators in L HD also affect Br 1.46 × 10 −3 . However, recall that when looking for the Higgs, the Z decay has to be multiplied by Br(Z → ).) We neglect these effects when fitting for the allowed parameter space. We also do not include the effects of these operators on h → W W, Z Z as the SM contribution is tree level for these processes. Further, we also neglect effects due to higher dimensional operators possibly modifying the differential distributions of the Higgs decay products, indirectly affecting theμ i through modifying the effective signal efficiency for specific kinematic cuts. Such effects are expected to be negligible compared to the current uncertainties. However, we do not neglect the rescaling effect on Γ h→gg that is identified with the rescaling on σ gg→h in Eq. 26. We include this rescaling consistently, which has a non-negligible impact on all branching ratios through the modification of Γ SM .
We show in Fig.12  or 0.5. In Fig.13 the residual χ 2 distribution for Br inv is shown whenc G ,c γ are marginalized over (left) and we also show the allowedc G ,c γ parameter space when Br inv is marginalized over (right) subject to the prior constraint 0 ≤ Br inv ≤ 1. These results show the significant impact of the higher dimensional operators, in scenarios consistent with the assumptions of this section, on attempts to extract Br inv from global fits to Higgs signal strength data.
Most notably, we find that the slight preference in the global χ 2 distribution for Br inv > 0 is removed when marginalizing over such unknown BSM effects in the current data set. This offers further caution to over interpreting the slight preference in the global χ 2 distribution for Br inv > 0 at this time. Although in Fig.12 the required Wilson coefficientc γ to still obtain a good fit when Br inv 0 is large, we find that even restrictingc G ,c γ to clearly perturbative couplings (≤ 1), expected in many models, the preference for a Br inv > 0 is removed. This result is shown in Fig.13 (left).

IV. PROSPECTS FOR DIRECT CONFIRMATION OF Br inv .
As stated above, there is a degeneracy between the case a = c < 1 and the existence of a non-zero invisible decay. The former leaves the branching ratios unchanged due to a common suppression factor in the couplings, while all production channels are suppressed by the same common factor. On the other hand, in the simple case of leaving the SM couplings unchanged but allowing for an invisible width, the production channels are unchanged and the branching ratios are affected as in Eq. 3, leading to a common overall suppression of production times branching ratio compared to the SM. If a signal is seen with suppressed event rate with respect to the SM expectation the degeneracy between these two cases can only be removed by observing directly a non-vanishing invisible decay.
It has been shown that associated production with gauge bosons, weak boson fusion and associated production with top quarks allows one to discover a Higgs boson decaying invisibly, and to probe the invisible branching ratio. The typical signature is large missing transverse energy/momentum. Assuming an invisible branching ratio of 1, a Higgs boson with mass up to about 150 GeV can be discovered in Higgs radiation from a Z boson at L = 10 fb −1 and √ s = 14 TeV [36][37][38][39][40][41][42][43]. At high luminosity this reach can be extended to ∼ 250 GeV in associated production with a top quark pair [43][44][45]. Weak boson fusion allows for the discovery up to 480 GeV with 10 fb −1 integrated luminosity [43,[46][47][48]. Assuming SM production, invisible branching ratios as low as 25% can be probed in weak boson fusion for a 120 GeV Higgs boson at L = 30 fb −1 and √ s = 14 TeV at 95% CL [46][47][48]. In associated production with a Z boson, branching ratios down to 45% can be probed [38,41,42] while associated Higgs production with a top quark pair probes invisible branching ratios down to 56% [45]. However, to claim a discovery of Br inv will require a combination of these global searches and direct kinematic searches, as we have demonstrated throughout Section III.
In this paper, we have systematically examined the potential of global fits to extract information The data we have used in the global fits of this paper are summarized in the table below.
Due to an apparent inconsistency in the ATLAS best fit signal strength plot for h → bb and the corresponding ATLAS CL s limit plot (that is under investigation by ATLAS) we do not use the bb best fit signal strength value in the combined fit at this time. For the pp → γ γ jj signal of CMS we assume a 3% contamination due to gg Higgs production events so that the relevant signal rate is given by (0.03 σ gg→h + σ jjh ) × Br(h → γ γ).
Here σ jjh is given by VBF Higgs production. We do not use sub classes of WW events due to the lack of experimentally reported contaminations of these signal strengths due to other Higgs production processes. Simultaneously using a global best fit valueμ for γ γ events (for example) while also using a best fitμ for a subclass of events, such as γ γ jj can result in a double counting of signal strengths that would incorrectly bias the fit. We avoid such double counting in our use of CMS and ATLAS data as the photon classes we use are exclusive, but note that double counting of this form is present in Ref. [21], making it difficult to compare results. In particular, to avoid introducing such a bias is why we use the experimentally reported global ATLASμ γγ , as a complete set of subchannel di-photon signal strengths is not available (in contrast to CMS). This is also the reason that we do not simultaneously use constructed signal strengthsμ γγ andμ γγ,P T >40Gev (from fermiophobic [54] searches). These signal strengths are not independent mutually exclusive event classes, being derived from the same signal event data. Our approach to this issue is different than the approach of Ref. [21].   was found to be stable against randomly chosen correlations. Also we have found in Section II consistent results between two different approaches to the fit of signal-strength parameters: using the individual channels or the combined results. This indicates that neglected correlations do not bias the fit results outside the quoted errors.
Note that here we use a value of 0.35 +1.08 −0.31 for the pp → W + W − [CDF&D0 /] result for m h = 124 GeV, unlike in Ref. [10], where we used the same value as for m h = 125 GeV. This introduces a small interpolation error, but allows better agreement with global combined signal strengths reported by the Tevatron collaboration.