NLO Higgs Effective Field Theory and kappa-framework

A consistent framework for studying Standard Model deviations is developed. It assumes that New Physics becomes relevant at some scale beyond the present experimental reach and uses the Effective Field Theory approach by adding higher-dimensional operators to the Standard Model Lagrangian and by computing relevant processes at the next-to-leading order, extending the original kappa-framework.


Introduction
During Run−1 LHC has discovered a resonance which is a candidate for the Higgs boson of the Standard Model (SM) [1,2]. The spin-0 nature of the resonance is well established [3] but there is no direct evidence for New Physics; furthermore, the available studies on the couplings of the resonance show compatibility with the Higgs boson of the SM. One possible scenario, in preparation for the results of Run−2, requires a consistent theory of SM deviations. Ongoing and near future experiments can achieve an estimated per mille sensitivity on precision Higgs and electroweak (EW) observables. This level of precision provides a window to indirectly explore the theory space of Beyond-the-SM (BSM) physics and place constraints on specific UV models. For this purpose, a consistent procedure of constructing SM deviations is clearly desirable.
The first attempt to build a framework for SM-deviations is represented by the so-called κ -framework, introduced in Refs. [4,5]. There is no need to repeat here the main argument, splitting and shifting different loop contributions in the amplitudes for Higgs-mediated processes. The κ -framework is an intuitive language which misses internal consistency when one moves beyond leading order (LO). As originally formulated, it violates gauge-invariance and unitarity. In a Quantum Field Theory (QFT) approach to a spontaneously broken theory, fermion masses and Yukawa couplings are deeply related and one cannot shift couplings while keeping masses fixed.
To be more specific the original framework has the following limitations: kinematics is not affected by κ -parameters, therefore the framework works at the level of total cross-sections, not for differential distributions; it is LO, partially accomodating factorizable QCD but not EW corrections; it is not QFT-compatible (ad-hoc variation of the SM parameters, violates gauge symmetry and unitarity).
However, the original κ -framework has one main virtue, to represent the first attempt towards a fully consistent QFT of SM deviations. The question is: can we make it fully consistent so that the experimental collaborations can simply upgrade their studies of the Higgs boson couplings? The answer is evidently yes, although the construction of a consistent theory of SM deviations (beyond LO) is far from trivial, especially from the technical point of view.
In this work we will reestablish that Effective Field Theory (EFT) can provide an adequate answer beyond LO. Furthermore, EFT represents the optimal approach towards Model Independence. Of course, there is no formulation that is completely model independent and EFT, as any other approach, is based on a given set of (well defined) assumptions. Working within this set we will show how to use EFT for building a framework for SM deviations, generalizing the work of Ref. [33]. A short version of our results, containing simple examples, was given in Ref. [34] and presented in [35,36].
In full generality we can distinguish a top-down approach (model dependent) and a bottom-up approach. The top-down approach is based on several steps. First one has to classify BSM models, possibly respecting custodial symmetry and decoupling, then the corresponding EFT can be constructed, e.g. via a covariant derivative expansion [37]. Once the EFT is derived one can construct (model by model) the corresponding SM deviations.
The bottom-up approach starts with the inclusion of a basis of dim = 6 operators and proceeds directly to the classification of SM deviations, possibly respecting the analytic structure of the SM amplitudes.
The Higgs EFT described and constructed in this work is based on several assumptions. We consider one Higgs doublet with a linear representation; this is flexible. We assume that there are no new "light" d.o.f. and decoupling of heavy d.o.f.; these are rigid assumptions. Absence of mass mixing of new heavy scalars with the SM Higgs doublet is also required.
We only work with dim = 6 operators. Therefore the scale Λ that characterizes the EFT cannot be too small, otherwise neglecting dim = 8 operators is not allowed. Furthermore, Λ cannot be too large, otherwise dim = 4 higher-order loops are more important than dim = 6 interference effects. It is worth noting that these statements do not imply an inconsistency of EFT. It only means that higher dimensional operators and/or higher order EW effects (e.g. Ref. [38]) must be included as well.
To summarize the strategy that will be described in this work we identify the following steps: start with EFT at a given order (here dim = 6 and NLO) and write any amplitude as a sum of κ -deformed SM sub-amplitudes (e.g. t, b and bosonic loops in H → γγ). Another sum of κ -deformed non-SM amplitudes is needed to complete the answer; at this point we can show that the κ -parameters are linear combinations of Wilson coefficients.
The rationale for this course of action is better understood in terms of a comparison between LEP and LHC. Physics is symmetry plus dynamics and symmetry is quintessential (gauge invariance etc.); however, symmetry without dynamics does not bring us this far. At LEP dynamics was the SM, unknowns were M H (α s (M Z ), . . . ); at LHC (post the discovery) unknowns are SM-deviations, dynamics? Specific BSM models are a choice but one would like to try also a model-independent approach. Instead of inventing unknown form factors we propose a decomposition where dynamics is controlled by dim = 4 amplitudes (with known analytical properties) and deviations (with a direct link to UV completion) are (constant) combinations of Wilson coefficients. Only the comparison with experimental data will allow us to judge the goodness of a proposal that, for us, is based on the belief that deviations need a SM basis.
On-shell studies at LHC will tell us a lot, off-shell ones will tell us (hopefully) much more [39][40][41][42][43]. If we run away from the H peak with a SM-deformed theory, up to some reasonable value s ≪ Λ 2 , we need to reproduce (deformed) SM low-energy effects, e.g. VV and tt thresholds. The BSM loops will remain unresolved (as SM loops are unresolved in the Fermi theory). That is why we need to expand the SM-deformations into a SM basis with the correct (low energy) behavior. If we stay in the neighbourhood of the peak any function will work, if we run away we have to know more of the analytical properties.
The outline of the paper is as follows: in Section 2 we introduce the EFT Lagrangian. In Section 3 we describe the various aspects of the calculation; in Section 4 we present details of the renormalization procedure, decays of the Higgs boson are described in Section 5, EW precision data in Section 6. Technical details, as well as the complete list of counterterms and amplitudes are given in several appendices.

The Lagrangian
In this Section we collect all definitions that are needed to write the Lagrangian defined by where L 4 is the SM Lagrangian [44] and a n i are arbitrary Wilson coefficients. Our EFT is defined by Eq.(1) and it is based on a number of assumptions: there is only one Higgs doublet (flexible), a linear realization is used (flexible), there are no new "light" d.o.f. and decoupling is assumed (rigid), the UV completion is weakly-coupled and renormalizable (flexible). Furthermore, neglecting dim = 8 operators and NNLO EW corrections implies the following range of applicability: 3 TeV < Λ < 5 TeV .
We can anticipate the strategy by saying that we are at the border of two HEP phases. A "predictive" phase: in any (strictly) renormalizable theory with n parameters one needs to match n data points, the (n + 1)th calculation is a prediction, e.g. as doable in the SM. A "fitting" (approximate predictive) phase: there are (N 6 +N 8 + · · · = ∞) renormalized Wilson coefficients that have to be fitted, e.g. measuring SM deformations due to a single O (6) insertion (N 6 is enough for per mille accuracy).

Conventions
We begin by considering the field-content of the Lagrangian. The scalar field Φ (with hypercharge 1/2) is defined by H is the custodial singlet in (2 L ⊗ 2 R ) = 1 ⊕ 3. Charge conjugation gives Φ c i = ε i j Φ * j , or The covariant derivative D µ is with g 1 = −s θ /c θ and where τ a are Pauli matrices while s θ (c θ ) is the sine(cosine) of the weak-mixing angle. Furthermore Here a, b, · · · = 1, . . . , 3. Furthermore, for the QCD part we introduce G a µν = ∂ µ g a ν − ∂ ν g a µ + g S f abc g b µ g c ν .
Here a, b, · · · = 1, . . . , 8 and the f are the SU(3) structure constants. Finally, we introduce fermions, and their covariant derivatives t 0 = − i 2 g 3 0 0 g 4 (11) with g i = −s θ /c θ λ i and The Standard Model Lagrangian is the sum of several terms: i.e., Yang-Mills, scalar, gauge-fixing, Faddeev-Popov ghosts and fermions. Furthermore, for a proper treatment of the neutral sector of the SM, we express g 0 in terms of the coupling constant g, where Γ is fixed by the request that the Z − A transition is zero at p 2 = 0, see Ref. [45]. The scalar Lagrangian is given by We will work in the β h -scheme of Ref. [45], where parameters are transformed according to the following equations: Furthermore, we introduce the Higgs VEV, v = √ 2 M/g, and fix β h order-by-order in perturbation theory by requiring 0 | H | 0 = 0. Here we follow the approach described in Refs. [45,46]. Table 1: List of dim = 6 operators, see Ref. [48], entering the renormalization procedure and the phenomenological applications described in this paper
We need matching of UV models onto EFT, order-by-order in a loop expansion. If L = {O While overcomplete sets (e.g. those derived without using equations of motion) are useful for cross-checking, a set that is not a basis (discarding a priori subsets of operators) is questionable, e.g. it is not closed under complete renormalization and may lead to violation of Ward-Slavnov-Taylor (WST) identities [67][68][69]. Finally, a basis is optimal insofar as it allows to write Feynman rules in arbitrary gauges. Our choice is given by In Table 1 we drop the superscript (6) and write the explicit correspondence with the operators of the so-called Warsaw basis, see Ref. [48]. We also introduce where u stands for a generic up-quark ({u , c , . . . }), d stands for a generic down-quark ({d , s , . . . }) and l for {e , µ , . . . }.

From the Lagrangian to the S -matrix
There are several technical points that deserve a careful treatment when constructing S -matrix elements from the Lagrangian of Eq. (17). We perform field and parameter redefinitions so that all kinetic and mass terms in the Lagrangian of Eq. (17) have the canonical normalization. First we define and β h is fixed, order-by-order, to have zero vacuum expectation value for the (properly normalized) Higgs field.
Particular care should be devoted in selecting the starting gauge-fixing Lagrangian. In order to reproduce the free SM Lagrangian (after redefinitions) we fix an arbitrary gauge, described by four ξ parameters, The full list of redefinitions is given in the following equations, where we have introduced R Λ = M 2 /Λ 2 . First the Lagrangian parameters, secondly, the fields: where X ± , Y Z and Y A are FP ghosts. Finally, the gauge parameters, normalized to one: We introduce a new coupling constant where G F is the Fermi coupling constant and derive the following solutions: where the ∆R i are given in Table 3. One could also write a more general relation where non diagonal terms start at O(g 2 ). In this way we could also require cancellation of the Z−A transition at O(g 6 ) but, in our experience, there is little to gain with this option. We have introduced the following combinations of Wilson coefficients: With our choice of reparametrization the final result can be written as follows: where {Φ} denotes the collection of fields and {p} the collection of parameters. In the following we will abandon the Φ, p notation since no confusion can arise.

Overview of the calculation
NLO EFT (dim = 6) is constructed according to the following scheme: each amplitude, e.g. H →| f , contains oneloop SM diagrams up to the relevant order in g, (tree) contact terms with one dim = 6 operator and one-loop diagrams with one dim = 6 operator insertion. Note that the latter contain also diagrams that do not have a counterpart in the SM (e.g. bubbles with 3 external lines). In full generality each amplitude is written as follows: where g is the SU(2) coupling constant and For each process the dim = 4 LO defines the value of N (e.g. N = 1 for H → VV, N = 3 for H → γγ etc.). Furthermore, N 6 = N for tree initiated processes and N − 2 for loop initiated ones. The full amplitude is obtained by inserting wave-function factors and finite renormalization counterterms. Renormalization makes UV finite all relevant, on-shell, S -matrix elements. It is made in two steps: first we introduce counterterms for fields and parameters. Counterterms are defined by We construct self-energies, Dyson resum them and require that all propagators are UV finite. In a second step we construct 3 -point (or higher) functions, check their O (4) -finiteness and remove the remaining O (6) UV divergences by mixing the Wilson coefficients W i : Renormalized Wilson coefficients are scale dependent and the logarithm of the scale can be resummed in terms of the LO coefficients of the anomalous dimension matrix [11].
Our aim is to discuss Higgs couplings and their SM deviations which requires precise definitions [71][72][73]: Definition The Higgs couplings can be extracted from Green's functions in well-defined kinematic limits, e.g. residue of the poles after extracting the parts which are 1P reducible. These are well-defined QFT objects, that we can probe both in production and in decays; from this perspective, VH production or vector-boson-fusion are on equal footing with gg fusion and Higgs decays. Therefore, the first step requires computing these residues which is the main result of this paper.
Every approach designed for studying SM deviations at LHC and beyond has to face a critical question: generally speaking, at LHC the EW core is embedded into a QCD environment, subject to large perturbative corrections and we expect considerable progress in the "evolution" of these corrections; the same considerations apply to PDFs. Therefore, does it make sense to 'fit" the EW core? Note that this is a general question which is not confined to our NLO approach.
In practice, our procedure is to write the answer in terms of SM deviations, i.e. the dynamical parts are dim = 4 and certain combinations of the deviation parameters will define the pseudo-observables (PO) to be fitted. Optimally, part of the factorizing QCD corrections could enter the PO definition. The suggested procedure requires the parametrization to be as general as possible, i.e. no a priori dropping of terms in the basis of operators. This will allow us to "reweight" the results when new (differential) K -factors become available; new input will touch only the dim = 4 components. PDFs changing is the most serious problem: at LEP the e + e − structure functions were known to very high accuracy (the effect was tested by using different QED radiators, differing by higher orders treatment); a change of PDFs at LHC will change the convolution and make the reweighting less simple, but still possible. For recent progress on the impact of QCD corrections within the EFT approach we quote Ref. [23].

Renormalization
There are several steps in the renormalization procedure. The orthodox approach to renormalization uses the language of "counterterms". It is worth noting that this is not a mandatory step, since one could write directly renormalization equations that connect the bare parameters of the Lagrangian to a set of data, skipping the introduction of intermediate renormalized quantities and avoiding any unnecessary reference to a given renormalization scheme.
In this approach, carried on at one loop in [74], no special attention is paid to individual Green functions, and one is mainly concerned with UV finiteness of S-matrix elements after the proper treatment of external legs in amputated Green functions, which greatly reduces the complexity of the calculation.
However, renormalization equations are usually organized through different building blocks, where gauge-boson selfenergies embed process-independent (universal) higher-order corrections and play a privileged role. Therefore, their structure has to be carefully analyzed, and the language of counterterms allows to disentangle UV overlapping divergences which show up at two loops.
In a renormalizable gauge theory, in fact, the UV poles of any Green function can be removed order-by-order in perturbation theory. In addition, the imaginary part of a Green function at a given order is fixed, through unitarity constraints, by the previous orders. Therefore, UV-subtraction terms have to be at most polynomials in the external momenta (in the following, "local" subtraction terms). Therefore, we will express our results using the language of counterterms: we promote bare quantities (parameters and fields) to renormalized ones and fix the counterterms at one loop in order to remove the UV poles.
Obviously, the absorption of UV divergences into local counterterms does not exhaust the renormalization procedure, because we have still to connect renormalized quantities to experimental data points, thus making the theory predictive.
In the remainder of this section we discuss renormalization constants for all parameters and fields. We introduce the following quantities where ε = 4 − d, d is the space-time dimension, γ E = 0.5772 is the Euler -Mascheroni constant and µ R is the renormalization scale. In Eq.(37) we have introduced an auxiliary mass µ which cancels in any UV-renormalized quantity; µ R cancels only after finite renormalization. Furthermore, x is positive definite. Only few functions are needed for renormalization purposes, where the finite part is where p 2 = −s and Λ 2 = λ s , m 2 1 , m 2 2 is the Källen lambda function. Furthermore we introduce with the choice of the EW scale, x = M 2 W , in Eq. (37). Technically speaking the renormalization program is complete only when UV poles are removed from all, off-shell, Green functions, something that is beyond the scope of this paper. Furthermore, we introduce UV decompositions also for Green functions: given a one-loop Green function with N external lines carrying Lorentz indices µ j , j = 1, . . . , N, we introduce form factors, Here the set K a , with a = 1, . . . , A, contains independent tensor structures made up of external momenta, Kroneckerdelta functions, elements of the Clifford algebra and Levi-Civita tensors. A large fraction of the form factors drops from the final answer when we make approximations, e.g. vector bosons couple only to conserved currents etc.
Requiring that all (off-shell) form factors (including external unphysical lines) are made UV finite by means of local counterterms implies working in the R ξ ξ -gauge, as shown (up to two loops in the SM) in Ref. [75].
A full generality is beyond the scope of this paper, we will limit ourselves to the usual 't Hooft-Feynman gauge and to those Green functions that are relevant for the phenomenological applications considered in this paper.

Tadpoles and transitions
We begin by considering the treatment of tadpoles: we fix β h , Eq. (21), such that 0 |H| 0 = 0 [45]. The solution is where we split according to the following equation (see Eq. (37)) The full result for the coefficients β (n) is given in Appendix A. The parameter Γ, defined in Eq. (14), is fixed by the request that the Z − A transition is zero at p 2 = 0; the corresponding expression is also reported in Appendix A.

H self-energy
The one-loop H self-energy is given by The bare H self-energy is decomposed as follows: Furthermore we introduce Σ (n) The full result for the H self-energy is given in Appendix B.

A self-energy
The one-loop A self-energy is given by where the Lorentz structure is specified by the tensor and p 2 = −s. Furthermore the bare Π AA is decomposed as follows: It is worth noting that the A−A transition satisfies a doubly-contracted Ward identity The full result for the A self-energy is given in Appendix B.

W, Z self-energies
The one-loop W, Z self-energies are given by where the form factors are decomposed according to We also introduce the residue of the UV pole and the finite part: etc. The full result for these self-energies is given in Appendix B. We also introduce

Z−A transition
The Z−A transition (up to one loop) is given by where we have included the term in the bare Lagrangian starting at O(g 6 ). The full result for the Z−A transition is given in Appendix B.

The fermion self-energy
The fermion self-energy is given by with a decomposition etc. The full result for the fermion self-energies (f = ν, l, u, d) is given in Appendix B.

Dyson resummed propagators
We will now present the Dyson resummed propagators for the electroweak gauge bosons. The function Π I i j represents the sum of all 1PI diagrams with two external boson fields, i and j, to all orders in perturbation theory (as usual, the external Born propagators are not to be included in the expression for Π I i j ). We write explicitly its Lorentz structure, where V indicates SM vector fields, and p µ is the incoming momentum of the vector boson. The full propagator for a field i which mixes with a field j via the function Π I i j is given by the perturbative series where k 0 = k n+1 = i, while for l = n + 1, k l can be i or j. ∆ ii is the Born propagator of the field i. We writē and refer to∆ ii as the resummed propagator. The quantity (Π ∆) ii is the sum of all the possible products of Born propagators and self-energies, starting with a 1PI self-energy Π I ii , or transition Π I i j , and ending with a propagator ∆ ii , such that each element of the sum cannot be obtained as a product of other elements in the sum.
In practice it is useful to define, as an auxiliary quantity, the "partially resummed" propagator for the field i,∆ ii , in which we resum only the proper 1PI self-energy insertions Π I ii , namely, If the particle i were not mixing with j through loops or two-leg vertex insertions,∆ ii would coincide with the resummed propagator∆ ii . Partially resummed propagators allow for a compact expression for (Π ∆) ii , so that the resummed propagator of the field i can be cast in the form We can also define a resummed propagator for the i-j transition. In this case there is no corresponding Born propagator, and the resummed one is given by the sum of all possible products of 1PI i and j self-energies, transitions, and Born propagators starting with ∆ ii and ending with ∆ j j . This sum can be simply expressed in the following compact form,
The corresponding partially resummed propagator iŝ We only consider the case where V couples to a conserved current; furthermore, we start by including one-particle irreducible (1PI) self-energies. Therefore the inverse propagators are defined as follows: • H partially resummed propagator is given by • A partially resummed propagator is given by • W partially resummed propagator is given by • Z partially resummed propagator is given by • Z−A transition is given by where S ZA µν is given in Eq. (56) and AZ δ µν + g 6 s dZ (6) AZ δ µν − a AZ dZ The expressions corresponding to dim = 6 are rather long and we found convenient to introduce linear combinations of Wilson coefficients, given in Eq.(88).
The results for dim = 6 simplify considerably if we neglect loop generated operators, for instance one obtains full factorization for Π AA (0), The rather long expressions with PTG and LG operator insertions are reported in Appendix D. Results in this section and in Appendix D refer to the expansion of the 1PI self-energies; inclusion of 1PR components amounts to the following replacements

Finite renormalization
The last step in one-loop renormalization is the connection between renormalized quantities and POs. Since all quantities at this stage are UV-free, we term it "finite renormalization". Note that the absorption of UV divergences into local counterterms is, to some extent, a trivial step; finite renormalization, instead, requires more attention. For example, beyond one loop one cannot use on-shell masses but only complex poles for all unstable particles [77,71]. Let us show some examples where the concept of an on-shell mass can be employed. Suppose that we renormalize a physical (pseudo-)observable F, where m is some renormalized mass. Consider two cases: a) two-loop corrections are not included and b) m appears at one and two loops in F 1L and F 2L but does not show up in the Born term F B . In these cases we can use the concept of an on-shell mass performing a finite mass renormalization at one loop. If m 0 is the bare mass for the field V we write where M OS is the on-shell mass and Σ is extracted from the required one-particle irreducible Green function; Eq.(94) is still meaningful (no dependence on gauge parameters) and will be used inside the result.
In the Complex Pole scheme we replace the conventional on-shell mass renormalization equation with the associated expression for the complex pole

Wave function renormalization
Let us summarize the various steps in renormalization. Consider the V propagator, assuming that V couples to conserved currents (in the following we will drop the label V). We have where M is the V bare mass. The procedure is as follows: we introduce UV counterterms for the field and its mass, and write the (finite) renormalization equation where M OS is the (on-shell) physical mass. After UV and finite renormalization we can write the following Taylor expansion: where g exp is defined in Eq.(99). The wave-function renormalization factor for the field Φ will be denoted by For fermion fields we use Eq. (77) and introduce Next we multiply spinors by the appropriate factors, i.e.
where the wave-function renormalization factors are obtained from Eq.(58) For illustration we present the H wave-function factor and we expand any function of s as follows: Explicit expressions for the wave-function factors are given in Appendix F.

Life and death of renormalization scale
Consider the A bare propagator where R (n) are the residues of the UV poles and L (n) are arbitrary coefficients of the scale-dependent logarithms. Furthermore, The renormalized propagator is Furthermore, we can write Σ ren Finite renormalization amounts to write Σ ren AA (s) = Π ren AA (s) s and to use s = 0 as subtraction point. Therefore, one can easily prove that including O (6) contribution. Therefore we may conclude that there is no µ R problem when a subtraction point is available. After discussing decays of the Higgs boson in Section 5 we will see that an additional step is needed in the renormalization procedure, i.e. mixing of the Wilson coefficients. At this point the scale dependence problem will surface again and renormalized Wilson coefficients become scale dependent.

Decays of the Higgs boson
In this Section we will present results for two-body decays of the Higgs boson while four-body decays will be included in a forthcoming publication. Our approach is based on the fact that renormalizing a theory must be a fully general procedure; only when this step is completed one may consider making approximations, e.g. neglecting the lepton masses, keeping only PTG terms etc. In particular, neglecting LG Wilson coefficients sensibly reduces the number of terms in any amplitude.
It is useful to introduce a more compact notation for Wilson coefficients, given in Table 4 and to use the following definition: Definition The PTG scenario: any amplitude computed at O(g n g 6 ) has a SM component of O(g n ) and two dim = 6 components: at O(g n−2 g 6 ) we allow both PTG and LG operator while at O(g n g 6 ) only PTG operators are included.

Loop-induced processes: H → γγ
The amplitude for the process H(P) → A µ (p 1 )A ν (p 2 ) can be written as where a AA = s θ c θ a φ WB + c 2 θ a φ B + s 2 θ a φ W and c θ = c ren θ etc. The last step in the UV-renormalization procedure requires a mixing among Wilson coefficients which cancels the remaining (dim = 6) parts. To this purpose we define The matrix dZ W is fixed by requiring cancellation of the residual UV poles and we obtain T (6) HAA ; div → T Inclusion of wave-function renormalization factors and of external leg factors (due to field redefinition described in Section 2.4) gives Finite renormalization requires writing c ren θ = c W 1 + g 2 ren 16 π 2 dZ c θ , where g 2 F = 4 √ 2 G F M 2 W and c W = M W /M Z . Another convenient way for writing the answer is the following: after renormalization we neglect all fermion masses but t, b and write where Wilson coefficients are those in Tab where Wilson coefficients are the renormalized ones. In the PTG scenario we only keep a t φ , a b φ , a φ D and a φ in Eq.(143).
The advantage of Eq.(141) is to establish a link between EFT and κ -language of Ref. [4], which has a validity restricted to LO. As a matter of fact Eq.(141) tells you that κ -factors can be introduced also at NLO level; they are combinations of Wilson coefficients but we have to extend the scheme with the inclusion of process dependent, non-factorizable, contributions.
Returning to the original convention for Wilson coefficients we derive the following result for the non-factorizable part of the amplitude: where

H → Zγ
The amplitude for H(P) → A µ (p 1 )Z ν (p 2 ) can be written as The result of Eq.(145) is fully general and can be used to prove WST identities. As far as the partial decay width is concerned only P 21 HAZ will be relevant, due to p · e(p) = 0 where e is the polarization vector. We start by considering the 1PI component of the amplitude and obtain where T is given by Furthermore we can write the following decomposition: Explicit expressions for T HAZ , T ,b HAZ will not be reported here. The 1PR component of the amplitude is given by where A µα off HAA denotes the off-shell H → AA amplitude. It is straightforward to derive i.e. the complete amplitude for H → AZ is proportional to T µν and, therefore, is transverse. UV renormalization requires the introduction of counterterms, Once again, the last step in the UV-renormalization procedure requires a mixing among Wilson coefficients, performed according to Eq.(136). We obtain Elements of the mixing matrix derived from the process H → AZ are given in Appendix G. After mixing the result of Eq.(154) becomes Inclusion of wave-function renormalization factors and of external leg factors (due to field redefinition, defined in Section 2.4) gives Finite renormalization is performed by using Eq.(140). To write the final answer it is convenient to define dim = 4 sub-amplitudes T I HAZ ; LO (I = W, t, b): they are given in Appendix I. Another convenient way for writing T ren HAZ is the following: 12,22 T nfc HAZ ; t, i W ren The factorizable part is defined in terms of κ -factors, see Eq.(142) In the PTG scenario we only keep a t φ , a b φ , a φ D and a φ in Eq.(159).
Returning to the original convention for Wilson coefficients we derive the following result for the non-factorizable part of the amplitude: where In the PTG scenario there are only 3 non-factorizable amplitudes for H → AZ, those proportional to a φ t V , a φ b V and a φ D . The full results are reported on Appendix H where (161)

H → ZZ
The amplitude for H(P) → Z µ (p 1 )Z ν (p 2 ) can be written as The result in Eq.(162) is fully general and can be used to prove WST identities. As far as the partial decay width is concerned only P 21 HZZ ≡ P HZZ will be relevant, due to p · e(p) = 0 where e is the polarization vector. Note that computing WST identities requires additional amplitudes, i.e. H → φ 0 γ and H → φ 0 φ 0 .
We discuss first the 1PI component of the process: as done before the form factors in Eq.(162) are decomposed as follows: It is easily seen that only D contains dim = 4 UV divergences. The 1PR component of the process involves the A−Z transition and it is given by where the H → AZ component is computed with off-shell A. The r.h.s. of Eq.(164) is expanded up to O(g 3 g 6 ) and we will use Complete, bare, amplitudes are constructed where the O(gg 6 ) components are: UV renormalization requires introduction of counterterms, where F = D, P and dZ i = dZ (4) i + g 6 dZ (6) i . We obtain HZZ ; div , The explicit expressions for F (6) HZZ ; div will not be reported here. The last step in UV-renormalization requires a mixing among Wilson coefficients, performed according to Eq.(136). After the removal of the remaining (dim = 6) UV parts we obtain Elements of the mixing matrix derived from H → ZZ are given in Appendix G. Inclusion of wave-function renormalization factors and of external leg factors (due to field redefinition, introduced in Section 2.4) gives Finite renormalization is performed by using Eq.(140). The process H → ZZ starts at O(g), therefore, the full set of counterterms must be included, not only the dim = 4 part, as we have done for the loop induced processes.
It is convenient to define NLO sub-amplitudes; however, to respect a factorization into t, b and bosonic components, we have to introduce the following quantities: where W Φ ; φ denotes the φ component of the Φ wave-function factor etc. Furthermore, ∑ gen implies summing over all fermions and all generations, while ∑ gen excludes t and b from the sum. We can now define κ -factors for the process, see Eq.(142): and obtain the final result for the amplitudes Here we have introduced and are given in Appendix I. Non-factorizable dim = 6 amplitudes are reported in Appendix H, using again Eq.(177).

H → WW
The derivation of the amplitude for H → WW follows closely the one for H → ZZ. There are two main differences, there are only 1PI contributions for H → WW and the the process shows an infrared (IR) component.
The IR part originates from two different sources. Vertex diagrams generate an IR C 0 function: where we have introduced The second source of IR behavior is found in the W wave-function factor: The lowest-order part of the amplitude is The O(gg 6 ) components of H → WW are: With their help we can isolate the IR part of the H → WW amplitude Having isolated the IR part of the amplitude we can repeat, step by step, the procedure developed in the previous sections. There is a non trivial aspect in the mixing of Wilson coefficients: the dim = 6 parts of H → AA, AZ and P HZZ,HWW contain UV divercences proportional to W 1,2,3 ; once renormalization is completed for H → AA, AZ and ZZ there is no freedom left and UV finiteness of H → WW must follow, proving closure of the dim = 6 basis with respect to renormalization.
We can now define κ -factors for H → WW, see Eq.(142). They are as follows: Next we obtain the final result for the amplitudes where we have introduced and are given in Appendix I. Non-factorizable dim = 6 amplitudes are reported in Appendix H, using again Eq.(189).

H → bb(τ + τ − ) and H → 4 l
These processes share the same level of complexity of H → ZZ(WW), including the presence of IR singularities. They will be discussed in details in a forthcoming publication.

ElectroWeak precision data
EFT is not confined to describe Higgs couplings and their SM deviations. It can be used to reformulate the constraints coming from electroweak precision data (EWPD), starting from the S, T and U parameters of Ref. [76] and including the full list of LEP pseudo-observables (PO).
There are several ways for incorporating EWPD: the preferred option, so far, is reducing (a priori) the number of dim = 6 operators. More generally, one could proceed by imposing penalty functions ω on the global LHC fit, that is functions defining an ω -penalized LS estimator for a set of global penalty parameters (perhaps using merit functions and the homotopy method). One could also consider using a Bayesian approach [78], with a flat prior for the parameters. Open questions are: one κ at the time? Fit first to the EWPD and then to H observables? Combination of both?
In the following we give a brief description of our procedure: from Eq.(48) and Eq.(84) we obtain >From Eq.(56) and Eq.(84) we obtain >From Eq.(52) and Eq.(84) we derive Similarly, we obtain The S, T and U parameters are defined in terms of (complete) self-energies at s = 0 and of their (first) derivatives. However, one has to be careful because the corresponding definition (see Ref. [76]) is given in the {α , G F , M Z } scheme, while we have adopted the more convenient {G F , M W , M Z } scheme. Working in the α-scheme has one advantage, the possibility of predicting the W (on-shell) mass. After UV renormalization and finite renormalization in the α -scheme we define M 2 W ; OS as the zero of the real part of the inverse W propagator and derive the effect of dim = 6 operators.

W mass
Working in the α -scheme we can predict M W . The solution is whereĉ 2 θ is defined in Eq.(109) and we drop the subscript OS (on-shell). Corrections are given in Appendix J. The expansion in Eq.(194) can be improved when working within the SM (dim = 4), see Ref. [44]: for instance, the expansion parameter is set to α(M Z ) instead of α(0), etc. Any equation that gives dim = 6 corrections to the SM result will always be understood as in order to match the TOPAZ0/Zfitter SM results when g 6 → 0, see Refs. [79][80][81] and Refs. [82,83].

S, T and U parameters
The S, U and T (the original ρ -parameter of Veltman [84]) are defined as follows:

Conclusions
In this paper we have developed a theory for Standard Model deviations based on the Effective Field Theory approach.
In particular, we have considered the introduction of dim = 6 operators and extended their application at the NLO level (for a very recent development see Ref. [85]).
The main result is represented by a consistent generalization of the LO κ -framework, currently used by ATLAS and CMS.
This step forward is better understood when comparing the present situation with the one at LEP; there the dynamics was fully described within the SM, with M H α s (M Z ), . . . as unknowns. Today, post the LHC discovery of a Hcandidate, unknowns are SM-deviations. This fact poses precise questions on the next level of dynamics. A specific BSM model is certainly a choice but one would like to try a more model independent approach.
The aim of this paper is to propose a decomposition where dynamics is controlled by dim = 4 amplitudes (with known analytical properties) and deviations (with a direct link to UV completions) are (constant) combinations of Wilson coefficients for dim = 6 operators.
Generalized κ -parameters form hyperplanes in the space of Wilson coefficients; each κ -plane describes (tangent) flat directions while normal directions are blind. Finally, κ -planes intersect, providing correlations among different processes. Our prescription allows a theoretically robust matching between theory and experiments.
Only the comparison with experimental data will allow us to judge the goodness of a proposal that, for us, is based on the belief that SM deviations need a SM basis.

A Appendix: β h and Γ
In this Appendix we present the full result for β h , defined in Eq. (43). We have introduced ratios of masses etc. The various components are given by We also present the full result for Γ, defined in Eq. (14). We have i.e. Γ = Γ (4) 1 + 2 g 6 a φ W .

B Appendix: Renormalized self-energies
In this Appendix we present the full set of renormalized self-energies. To keep the notation as compact as possible a number of auxiliary quantities has been introduced.

B.1 Notations
First we define the following set of polynomials:

B.2 Renormalized self-energies
The (renormalized) bosonic self-energies are decomposed according to while the (renormalized) fermionic self-energies are decomposed as In the following list we introduce a shorthand notation: and several linear combinations of Wilson coefficients All functions in the following list are decomposed according to where F 0 is the constant part (containing a dependence on µ R ), F 1 contains (finite) one-point functions and F 2 the (finite) two-point functions. Capital letters (U etc.) denote a specific fermion, small letters (u etc.) are used when summing over fermions.

C Appendix: Listing the counterterms
In this Appendix we give the full list of counterterms, dropping the ren -subscript for the parameters, s θ = s θ ren etc. To keep the notation as compact as possible a number of auxiliary quantities are introduced. First we define the following set of polynomials: gen C a 12 = 9 + 4 v (2) gen + v (1) gen C a 13 = 17 + v (3) gen − 2 v (2) gen D where we have introduced gen − x (1) 1 c 2 (222)

C.2 dim = 6 counterterms
To present dim = 6 counterterms we define vectors; for counterterms and for Wilson coefficients etc. In the following we introduce all non-zero entries of the matrix M ct .

E Appendix: Finite counterterms
In this Appendix we present the list of finite counterterms for fields and parameters, as defined in Section 4.12. It should be understood that only the real part of the loop functions has to be included, i.e. B 0 ≡ Re B 0 etc. dZ (4)

F Appendix: Wave-function factors
In this Appendix we present the full list of wave-function renormalization factors for H, Z and W fields. For the W factor we present only the IR finite part.         In this Appendix we present the entries of the mixing matrix, Eq.(136), that can be derived from the renormalization of H → VV.

H Appendix: Non-factorizable amplitudes
In this Appendix we present the explicit expressions for the non-factorizable part of the H → AA, AZ, ZZ and WW amplitudes.

H.1 Notations
It is useful to introduce the following sets of polynomials: T where s = s θ and c = c θ