Higgs At Last

We update the experimental constraints on the parameters of the Higgs effective Lagrangian. We combine the most recent LHC Higgs data in all available search channels with electroweak precision observables from SLC, LEP-1, LEP-2, and the Tevatron. Overall, the data are perfectly consistent with the 126 GeV particle being the Standard Model Higgs boson. The Higgs coupling to W and Z bosons relative to the Standard Model one is constrained in the range [0.98,1.09] at 95% confidence level, independently of the values of other Higgs couplings. Higher-order Higgs couplings to electroweak gauge bosons are also well constrained by a combination of LHC Higgs data and electroweak precision tests.


Introduction
The existence of a boson with a mass around m h = 126 GeV is firmly established [1]. The focus now is on determining the properties of the new particle, in particular its couplings to the Standard Model (SM) matter. If new physics beyond the SM plays a role in electroweak (EW) symmetry breaking, then Higgs couplings may be modified in an observable way. So far, the measured couplings of the new particle are consistent with those of the SM Higgs boson; nevertheless, the current experimental precision leaves ample room for new physics.
A general framework to study potential deviations of Higgs couplings from the SM is that of an effective theory. The basic assumption behind this approach is that new degrees of freedom coupled to the Higgs are heavy enough such that their effects can be described by means of local operators involving the SM fields only. These operators can be organized into a formal derivative expansion, according to the relevance for Higgs observables: the leading order (LO) operators with no derivatives, the next-to-leading order (NLO) operators suppressed by two derivatives, and so on. Previous studies along these lines have shown that the coefficients of the leading operators in this expansion can already be meaningfully constrained [2,3].
The purpose of this paper is two-fold. One is to update the constraints on the effective theory parameters using the most recent Higgs data from the LHC [4]- [16]. The other is to point out that not only the LO but also some NLO operators in the effective theory can be meaningfully constrained using the existing data. This can be achieved by combining the LHC Higgs data and EW precision constraints. In this paper we concentrate on the class of models that are favored by EW precision data and study the constraints from SLC, LEP-1, LEP-2 and W mass measurement on the Higgs couplings, including the very recent update of the fermionic cross section measurements at LEP-2 [17]. We will argue that Higgs and electroweak data taken together impose strong bounds on the coefficients of the LO and NLO operators coupling the Higgs boson to the SM gauge bosons.
The paper is organized as follows. In Sec. 2 we introduce a general Higgs low-energy effective Lagrangian, subject only to minimal assumptions about flavor and some EW precision constraints. In Sec. 3 we further discuss the bounds on the parameters of our Lagrangian from electroweak precision tests; to this end we perform a comprehensive and up-to-date In complete generality, we don't assume that the scalar h originates, as in the SM, from a fundamental SU (2) L doublet scalar field. Our description equally applies if h descends from different representations under the electroweak gauge group (singlet, triplets, etc. ), or is not fundamental (composite Higgs, dilaton, etc.). Nevertheless, with only a slight abuse of language, we shall refer to it as a Higgs boson.
In this paper we assume that: • Near 126 GeV there is a unique Higgs boson who is a color-singlet neutral scalar with positive parity.
• This Higgs has naturally no flavor violating interactions with the SM fermions.
• Higgs interactions obey custodial symmetry under which h is a singlet.
With these assumptions, the lowest-order interaction Lagrangian takes the form: As a consequence of custodial symmetry, only one parameter c V controls the LO couplings to both W and Z bosons; relaxing this would lead to quadratically divergent corrections to the T parameter, and thus any departure from custodial symmetry is severely constrained at the level of 1%, barring large fine-tuned cancellations. Furthermore, while we allow the couplings to up-type quarks, down-type quarks, and leptons to be independent, we assume that within each of these classes the coupling ratios are equal to the fermion mass ratio. Relaxing this would generically lead to flavor-changing Higgs interactions in the mass eigenstate basis, which are very constrained unless some underlying flavor principle is at work to suppress these dangerous effects.
At the NLO in the derivative expansion we include where custodial symmetry imposes two further restrictions on the couplings (see Appendix A), Thus, a combination of reasonable assumptions, effective theory arguments, and phenomenological constraints leads us to the effective Higgs interaction Lagrangian with only 7 free parameters: In the remainder of this paper we discuss how data constrains these parameters.

Electroweak Precision Tests
It is well known that electroweak precision observables are sensitive to the mass and couplings of the Higgs boson. At the technical level, the largest effect enters via 1-loop contributions to the 2-point functions of the electroweak gauge bosons, the so-called oblique corrections. Once the single Higgs couplings deviate from the SM, the sum of the Higgs and pure gauge contributions to precision observables becomes divergent. Indeed, as discussed in Appendix A, custodial breaking parameters such as c z V − c w V (and at NLO couplings such as hZ µ ∂ ν V µν ) would give too large a contribution to the EW precision parameters and have been neglected in our Lagrangian Eq. (2.2) and Eq. (2.3). Furthermore we assume that other contributions to the electroweak precision parameters, such as tree-level effects or loop effects involving only gauge bosons, are negligible (the presence of such contributions would weaken the connection between EW precision tests and LHC data that we assume here). Therefore, in our Lagrangian, the divergences are only logarithmic, thanks to custodial symmetry and forbidding this class of operators. As for the possible h 2 interactions with the gauge bosons (which we ignore in this paper), we argue here that in a natural situation they do not play a role. Effective operators involving two Higgs bosons of the type h 2 V µ V µ , or h 2 V µν V µν give a contribution to the gauge bosons two-point functions that is always proportional to the com- . Therefore, imposing the cancellation of quadratic divergences always cancels the logarithmic terms as well. Effective operators of the type h(∂ µ h)V µ do not give any contribution at all. Thus, barring an unnatural situation with quadratically divergent contributions to precision observables, it is enough to consider the electroweak constraint on the parameters c V , c γγ , c Zγ in our effective Lagrangian.
It is instructive to view the 1-loop Higgs contributions to the oblique parameters S, T , where we have omitted finite contributions in the leading-log approximation (see Ref. [23] where these contributions, including large tree-level effects, have been computed in an explicit In Fig. 1 we show two examples of electroweak precision constraints on the parameters of the effective Lagrangian. 3 We use the data from SLC, LEP-1, LEP-2 and W mass 2 See Appendix B for the definition and a discussion of these parameters. The more familiar S, T, U parametrization is not adequate in our case, since U is not logarithmically enhanced, unlike W and Y . 3 See also [3,21] for previous studies of the impact of electroweak precision tests in the context of the 126 GeV Higgs.    Table 3.4 Ref. [17] [28] µ + µ − forward-backward asym. at LEP-2 Table 3.4 Ref. [17] [28] τ + τ − forward-backward asym. at LEP-2 Table 8.9 Ref. [26] [28] σ(e + e − → bb)/σ(e + e − → qq) at LEP-2 R c Table 8.10 Ref.
[26] for heavy flavor observables in LEP-2, and Ref. [17] for the remaining LEP-2 observables. To obtain the SM predictions we used Gfitter for the W mass and Z-pole observables [27], and ZFITTER [28] for LEP-2 observables except for the e + e − → e + e − differential cross section for which we used BHWIDE [29]. The input parameters are In this section we summarize how the LHC Higgs observables depend on the parameters of our effective Lagrangian. As customary, we present the results in the form of rates in various channels relative to the SM ones.

Decay
The widths for fermionic decays mediated by the LO couplings Eq. (2.2) are given by Notice that we neglect the NLO couplings in loop diagrams, as these are formally higherorder contributions.
The decay amplitudes for the processes h → 4l and h → 2l2ν via intermediate ZZ Integrating the amplitude squared over the 4-body final state phase space we obtain Apparently, the contribution from the NLO Higgs couplings to these decays is suppressed by the off-shellness of the intermediate gauge boson(s), thus for c γγ and c Zγ of natural size their effect on the h → V V * decay widths is expected to be negligible. 4 Finally, we recall that the branching fraction in a given channel is the total decay width is now given by

Production Cross Sections
Given the current experimental data, the important partonic processes for Higgs production, and their relative cross sections in terms of the effective theory parameters are the following.

Results and discussion
In this section we present a fully up-to-date fit to the parameters of the Higgs effective Lagrangian. We combine the available Higgs results summarized in Table 2 and EW precision observables from Table 1. Throughout this section we assume Λ = 3 TeV to estimate the logarithmically divergent Higgs contributions to oblique parameters.

7-parameter fit
We begin with an unconstrained fit to all seven parameters of the effective Lagrangian. We When quoting 1σ errors above we ignored other isolated minima away from the SM point where a large NLO coupling conspires with the SM loops to produce a small shift of the Higgs observables. We find χ 2 SM − χ 2 min = 4.2 which means that the SM gives a perfect fit to the Higgs and EW data. The previous small discrepancy due to the enhanced h → γγ rate observed by the ATLAS and CMS goes away after including the latest CMS update in the diphoton channel. Remarkably, the current data already puts meaningful limits on all the parameters. The only important degeneracy is that between c gg and c t : only one linear combination of these two is constrained by stringent limits on the gluon fusion Higgs production, while direct constraints on c t from searches of the tt associated Higgs production are currently weak. Note that the global fit shows a strong preference for c b = 0 even though

Categoryμ
Ref.  As is clear from Fig. 1, c V > 0 with enormous statistical significance.
Regarding the NLO couplings, it must be recalled that these coefficient are expected to arise at the loop level and to be suppressed by the SM gauge couplings. With the appropriate rescalings (πα s , πα and s w πα, see Appendix A), we find that the values above correspond to coefficients of a gg, γγ and Zγ, of order 0.3 15 and 6, respectively. It is interesting to notice, by comparing Figs. 1 and 2, that the strongest constraints on the c γγ coupling already come from the LHC rather than from LEP.

Loop New Physics
We Also in this more constrained case the SM point c gg = c γγ = c Zγ = 0 has χ 2 SM − χ 2 min = 2.6, and thus provides a perfect fit to the data. Once the degeneracy with c t is removed, the constraints on c gg are at the level 10 −3 , much as the constraints on c γγ . The remaining NLO coupling c Zγ is less constrained, at the level of few×10 −2 , because, for the time being, the decay h → Zγ is not observed. When setting the 1σ errors in Eq. (5.2) we ignored other degenerate minima of χ 2 away from the SM-like minimum; we comment on them below.
In Fig. 2 we also show two sections of the loop-inspired parameter subspace. In the left panel of Fig. 2, we vary c gg and c γγ , while the least constrained coupling c Zγ is set to zero for simplicity. In this case, the EW precision observables have little relevance, as they do not depend on c gg at one loop, while the Higgs constraints on c γγ are far more stringent. If the NLO couplings are induced by a loop of a single new particle with charge Q and color representation r then it follows c γγ = − d(r)Q 2 C 2 (r) α αs c gg . For example, a fermionic or bosonic particle in the same color representation as SM quarks has C 2 (r) = 1/2 and d(r) = 3. In In the right panel of Fig. 2 we vary c γγ and c Zγ and set c gg = 0, which is relevant for integrating out colorless particles. The constraints on c Zγ from EW precision observables and from Higgs data are competitive. In contrast to the previous case, there are only 2 best-fit regions due the fact there's currently no evidence of the h → Zγ decay. The h → Zγ rate relative to the SM can be enhanced by a factor of a few, but it could also be suppressed; the Higgs and EW data show no strong preference for either of these possibilities. We also show the contours of W H associated production cross section relative to the SM one. An enhancement up to 40% of W H and ZH production due to NLO Higgs couplings is still perfectly consistent with the current data. Besides, the relative change of the W H and ZH production cross section can be different by up to 5%, even though custodial symmetry is preserved.

Composite Higgs
Here we present the results of the 2-parameter fit under the assumption that all the Higgs couplings to fermions take a common value c t = c b = c τ ≡ c f . The other free parameter in this model is the LO Higgs coupling to W and Z bosons c V . The NLO couplings in the effective Higgs Lagrangian are all set to zero. This subspace of the parameter space is inspired by the physics of composite Higgs where a light Higgs boson arises as a pseudo-Goldstone boson of a spontaneously broken approximate global symmetry in a strongly interacting sector [34]. In this class of models, the Higgs coupling to W and Z is suppressed where n, m are positive integers. The models with m = 0 are already excluded by the data, while m = 0, n = 0 corresponds to the MCHM4 model and m = 0, n = 1 to the MCHM5 model [36]. Finally, the NLO Higgs couplings of the form |H| 2 V 2 µν that could give rise to non-zero c γγ , c gg in our effective Lagrangian are expected to be suppressed because they violate the shift symmetry of the pseudo-Goldstone boson Higgs.
The results of the fit are given in the left panel of Fig. 3. The island of good fit for c f < 0 favored by previous Higgs data [2,3] completely vanishes in the new data. The reason is that a large enhancement of the h → γγ rate is no longer preferred. The preference for the SM-like coupling c f ∼ 1 becomes even stronger when EW precision data are included. This is true in spite of the fact that the EW observables are not sensitive to c f at one loop; simply, they prefer c V somewhat above 1 which is more consistent with the SM island. In the right panel of Fig. 3 we show the bounds on the compositeness scale for a number of composite Higgs models based on the SO(5)/SO(4) coset. We find that the strongest bounds still come from EW precision data and, as already pointed out in Ref. [37], they push the compositeness scale f at about 1.5 TeV at 95% CL, independently of the specific model. Nevertheless, incalculable UV effects could weaken the impact of EW precision data; in this case, and taking into account Higgs data only, the bound on f reduces to the more natural value f 700 GeV, with some dependence on the details of the model.

2HDM
Another interesting pattern of couplings is the one where the Higgs couplings to leptons and down-type quarks take a common value c d ≡ c b = c τ which differs from the coupling to up-type quarks c u ≡ c t . At the same time, the LO coupling to gauge bosons doesn't deviate from the SM, c V = 1, and NLO couplings vanish. Notice that this plane is insensitive to bounds from EW precision data. This parametrization is inspired by type-II two-Higgs doublets models (2HDM), in particular by minimal supersymmetry. Indeed, in supersymmetric models, assuming heavy superpartners as presently suggested by experiments, the peculiar they reduce to [38,39], where tan β > 1 and δ depends on the underlying physics that contributes to the Higgs mass (for instance δ > 0 in the MSSM and in models with additional D-terms, while δ < 0 in the NMSSM) [39]. Corrections to c V arise at higher order in the v/m H expansion and are typically very small.
We show the fit to c b and c t in Fig. 4. SUSY models imply deviations which lie in the upper-left or lower-right quadrant unless scalar singlets mix with the Higgs. The best-fit region at negative c t preferred in the previous fits [38,39] is now only marginally allowed after including the latest CMS results in the γγ channel [4].

Invisible Width
In the last of our studies we are going beyond our effective Lagrangian and allow Higgs decays to invisible particles. Searches for such decays are strongly motivated by the so-called Higgs portal models of dark matter (see Ref. [40] for a natural realization of this scenario).
Furthermore, given the small Higgs width in the SM, Γ H,SM ∼ 10 −5 m h , a significant invisible width Γ H,inv ∼ Γ H,SM may easily arise even from small couplings of the Higgs to weakly interacting new physics. In supersymmetric models where the Higgs is the superpartner of a slepton [41], there could be invisible decays into neutrinos and gravitinos.
If the Higgs couplings to the SM matter take the SM values, then the invisible width leads to a universal reduction of the rates in all visible channels. This possibility is already quite constrained, given that we do see the Higgs produced with roughly the SM rate. The left panel of Fig. 5 shows ∆χ 2 as a function of the invisible branching fraction: Br inv > 22% is excluded at 95% CL. 5 This bound can be relaxed if new physics modifies also the Higgs 5 Simply combining the overall Higgs signal strengths quoted by ATLAS µ = 1.30 ± 0.20 [16], CMS µ = 0.80 ± 0.14 [8], and Tevatron µ = 1.44 +0.59 −0.56 [33] one obtains the bound Br inv > 23.6% at 95% CL, fairly close to the result of our fit. couplings such that the Higgs production cross section is enhanced. An example of such set-up is plotted in the right panel of Fig. 5, where we show the allowed region assuming the invisible Higgs branching fraction and, simultaneously, a non-zero NLO coupling to gluons.
Even in this more general case Br inv larger than ∼ 50% is excluded at 95% CL. The indirect limits on the invisible width are in most cases much stronger than the direct ones from the ATLAS Z + h → inv. search and from monojet searches. [42], and monojet constraints (black) derived in [43] using the CMS monojet search [44].

Conclusions
In this paper we updated the experimental constraints on the parameters of the Higgs effective Lagrangian. We combined the most recent LHC Higgs data in all available channels with the electroweak precision observables from SLC, LEP-1, LEP-2, and the Tevatron. Overall, the data are perfectly consistent with the 126 GeV particle discovered at the LHC being the SM Higgs boson. A slight tension with the SM and a preference for negative Yukawa couplings found in previous Higgs fits goes away after including the latest CMS data in the h → γγ channel. The leading order Higgs couplings to the SM matter in Eq. (2.2) are well constrained, especially the coupling c V to W and Z bosons which is constrained by a combination of Higgs and EW data to be within 10% of the SM value at 95% CL.
The corollary is that the new particle is a Higgs boson: it couples to the mass of the W and Z bosons, therefore it plays a role in electroweak symmetry breaking. This statement is independent of the size of possible higher-order Higgs couplings to W and Z (that play no role in electroweak symmetry breaking). Higher-order (2-derivative) couplings in the ef- manifest (but non-linearly realized). We then show how this relates to the phenomenological Lagrangian of Eqs. (2.2,2.3) and to the conditions Eq. (2.4) and classify the operators according to their expected size in a wide class of theories, in the spirit of Ref. [18]. While the latter discusses a strongly interacting light Higgs doublet (SILH), their arguments, which we follow closely in what follows, are useful also in the weakly coupled regime [45].
The goldstone boson matrix U ≡ exp(iϕ j τ j /v) has well defined transformation properties and is the building-block for the Lagrangian with broken (non-linearly realized) EW symmetry. Furthermore, transform as adjoints of SU (2) L and singlets of The most general Lagrangian describing gauge bosons and h can be written as an expansion in 6 where B µν is the usual U (1) Y field-strength tensor, D µ the covariant derivative and g h is the coupling of h to the sector responsible to generate the corresponding term in the Lagrangian. In composite Higgs models, where h arises as a resonance from a strong sector whose dynamics breaks the EW symmetry, we expect g h v ≈ Λ (for simplicity we shall assume g h v/Λ = 1 in what follows, while we remark that the most general case can be obtained by accompanying each insertion of h with a factor g h v/Λ).
We are interested in operators up to dimension 5, involving two gauge bosons and one scalar h, as these affect both EW precision tests and the h decay widths. At leading order in the derivative expansion, the only terms are of operator O i in the Lagrangian. O C 1 breaks custodial symmetry even in the g Y → 0 limit and is absent in Eq. (2.2). It is instructive to compare Eq. (A.4) with the analogous results of Ref. [18], which makes the additional assumption that h is part of an SU (2) L doublet that breaks the EW symmetry: we find the contribution to the above operators Notice that, although the two contributions coincide in the limit f → v, in realistic theories one finds f v.
At the next order in the derivative expansion we have several contributions, which we classify accordingly to their expected size. The operators 7 can arise at tree level mediated by a vector with mass Λ and the appropriate quantum numbers [48]. They generate structures like in Ref. [18].
7 Other combinations can be eliminated using the identity In particular Tr[T (D µŴ µν )V ν ] is a linear combination of O 2 , O C 2 and terms with more than two gauge bosons.

The operators
are instead expected to arise at the loop level in minimally coupled theories (i.e. theories where gauge bosons couple only through covariant derivatives [18,48,45]). These Notice that, if the Higgs doublet arises as a pseudo Nambu-Goldstone boson of a strongly interacting sector [34], then O γ is further suppressed by ∼ g 2 /g 2 h , implying a smaller c γγ in Eq. (2.3).

Finally the operators
complete our list 8 . These terms are in principle expected to be larger than the ones in Eq. (A.7), coinciding only in the limit g h ∼ Λ/v → 4π. Nevertheless it is interesting to note that, if h is part of an SU (2) L doublet and if we assume that the sector responsible for generating these couplings only involves particles with spin ≤ 1, then contribution to these terms do not arise at dimension 6, but rather at dimension 8 and are therefore suppressed by further powers of g h v/Λ = v/f which is are generally small [18]. This is also what is expected from a phenomenological point of view (they generate structures like (h/v)∂ µ Z ν ∂ ν Z µ ), since they contribute to the EW precision observables via quadratic divergences.
In implies in principle the existence of operators Tr[T (D µŴ µν )V ν ] and ∂ µ B µν Tr(T V ν ), which contribute at tree level to the EW precision parameters and are forced to be small. We are therefore assuming that the same dynamical mechanism that forbids tree-level effects, also accounts for the suppression of h-loop mediated effects.

B Oblique Parameters
The term oblique corrections refers to modifications of the propagators of the electroweak gauge bosons. In many models beyond the SM the largest new contributions to physical observables enter via the oblique corrections (for a more general approach to electroweak data, see [50]). Let us define an expansion of the 2-point functions in powers of p 2 , where p is the 4-momentum flowing through the diagram: where δΠ denotes a shift of the corresponding 2-point function from the SM value, and the fine-structure constant α ≈ 1/137 is used for normalization. At order p 4 one can define further oblique parameters [22]. 9 αV = m 2 W δΠ LEP-1 and SLC measurements on the Z resonance constrain two linear combinations of the 2 point-functions, and the W mass measurement constrains another. Thus, by itself, the above measurement can constrain only three obliques parameters, e.g. S, T and U , which explains the origin of the Peskin-Takeuchi parametrization. Adding the constraints from off-Z-pole measurements in LEP-2 allows one to put meaningful limits on V , W , X, and Y as well. Out of these seven oblique parameters four are singled out because, assuming the Higgs field is an SU (2) L doublet H, they correspond to dimension-6 operators beyond the SM. The mapping is whereas U , V , X correspond to operators of dimension 8 and higher. For this reason, typical new physics models affect S, T , W , Y in the first place. Combining the EW observables listed in Table 1 for U = V = X = 0 we obtain the following constraints: 9 Compared to Ref. [22] we rescaled these parameters by α so as to put them numerically on equal footing with S, T , U .