Scaling and tuning of EW and Higgs observables

We study deformations of the SM via higher dimensional operators. In particular, we explicitly calculate the one-loop anomalous dimension matrix for 13 bosonic dimension-6 operators relevant for electroweak and Higgs physics. These scaling equations allow us to derive RG-induced bounds, stronger than the direct constraints, on a universal shift of the Higgs couplings and some anomalous triple gauge couplings by assuming no tuning at the scale of new physics, i.e. by requiring that their individual contributions to the running of other severely constrained observables, like the electroweak oblique parameters or $\Gamma(h \rightarrow \gamma\gamma)$, do not exceed their experimental direct bounds. We also study operators involving the Higgs and gluon fields.


Introduction
The first run of the LHC, with the discovery of the Higgs boson and the measurement of its characteristic properties [1,2], has been extremely successful. There are, however, still no signs of any new physics that can stabilize the Higgs mass and thus a deeper understanding of the electroweak (EW) symmetry breaking sector of the Standard Model (SM) is still missing. All natural explanations for the EW symmetry breaking introduce new particles around the TeV scale which, when integrated out, alter the Higgs properties. Hence, measuring the Higgs sector with great accuracy has the potential of clarifying the origin of EW symmetry breaking. Assuming a mass gap between the SM scale and the new physics scale, as the lack of evidence for new physics seems to suggest, the Higgs properties and its deviations from the SM can be conveniently parametrized and systematically studied by higher dimensional operators [3], where the c i 's are dubbed Wilson coefficients.
While these higher dimensional operators are generated at the new physics scale Λ, they are measured at the lower scale of the experiments. 1 Due to renormalization group (RG) flow, the Wilson coefficients run and mix as we go down from Λ to the experimental scale ∼ m W . The operator coefficients at the two different scales are related to each other via the so-called anomalous dimension matrix. These quantum corrections mix the operators among themselves and therefore open the possibility of linking different kinds of deformations from the SM which are otherwise unrelated. In this article, we compute the anomalous dimension matrix of a set of 13 dimension-6 (dim-6) operators composed only of gauge bosons and Higgs fields and estimate the impact of these RG mixing effects on experimental measurements. Some elements of the anomalous dimension matrix have been previously calculated in the literature, see refs. [4][5][6][7][8][9][10][11][12][13][14], with a renewed interest after the recent Higgs property measurements.
To be completely general about the possible new physics scenarios one would need to compute the anomalous dimension matrix for all the 59 dim-6 operators [3,15]. 2 A given set of experimental observables, however, receives contributions only from a subset of these operators. The dim-6 operators we are focussing our attention on, is a particularly interesting subset as they capture most of the possible deformations of the electroweak sector studied at LEP (i.e., electroweak precision tests and triple gauge couplings) and of the Higgs sector being currently studied at the LHC. At the same time, these operators are among the most important ones generated by universal new physics theories. 3 See for instance refs. [16,17] and refs. [12,18] for a recent general phenomenological analysis of the SM operators; the last two stress the presence of blind directions on certain combinations of the Wilson coefficients [19].
One may naively think that these RG effects do not have a significant impact on phenomenology since they are loop suppressed. This is, however, not the case because the different Wilson coefficients have been constrained at different levels of precision. In particular, the ones contributing to LEP electroweak precision observables have been measured at the per mille level, whereas those parametrizing triple gauge couplings (TGC) and Higgs coupling data have been measured at most at the percent level. This hierarchy in the size of constraints means that, despite the one loop factor, the RG contributions of a weakly constrained coupling to a strongly constrained one can be of the same order as, or even larger than, the bound on the strongly constrained coefficient. This means that the RG-mixing effects of such weakly constrained Wilson coefficients can be measured/constrained by precision measurements of other couplings to which experiments are more sensitive. Indeed, we find interesting instances of coefficients which receive stronger bounds from the RG mixing than from the direct tree-level constraint. For example, we show that the Wilson coefficients parametrizing deviations in some of the anomalous TGC observables and the correction to the Higgs kinetic termĉ H receive a stronger bound via their RG-mixing contribution to the electroweak parametersŜ,T , W , Y and Γ h→γγ than the direct constraint. In refs. [6,8], and more recently refs. [20,21], the RG effects of the mixing of TGC and the EW parameters are studied.
The paper is organized as follows. In Section 2 we define the basis of dim-6 operators we shall use. Then, in Section 3 we present our result for the anomalous dimension matrix of the 10 bosonic 4 operators related to EW and Higgs observables. In Section 4 we shall use the RG equations (RGE's) to set bounds on the value of some Wilson coefficients that are otherwise less constrained by direct measurements; we also comment on the future prospects. In Section 5 we present the anomalous dimension matrix for a set of operators with gluons and discuss the available bounds on them. We conclude in Section 6. In the appendices, we report several details of our computations and present a comparison with existing results in literature.

The dimension-six operator basis
In this section we define our choice for the dim-6 operator basis [3,15] and the subset of the dim-6 operators for which we want to compute the anomalous dimension submatrix. Our choice of basis is motivated by the observables we are interested in, and the subset we consider is defined by the operators in this basis which give a tree-level contribution to our set of observables. In this work, we shall be interested in EW observables, Higgs couplings to gauge bosons and QCD observables involving gluons only and the relations among each other as imposed from the running between the scale of new physics to the weak scale. These include the four electroweak oblique pseudo-observablesŜ,T , W and Y , the three triple gauge coupling observables g Z 1 , κ γ and λ γ , the Higgs couplings to vector bosons, the gluon oblique parameter Z [22] and the anomalous triple gluon coupling parameterĉ 3G . We describe these observables in more detail in Section 4.2 and Section 5. For ealier systematic studies of the effects of higher-dimensional operators on these observables, see refs. [23,24].
We have not included the Higgs decays to fermions in our list of observables. The only dim-6 operators contributing to these observables are the operators O yu , O y d and O ye , defined in ref. [12]. 5 The RG effects of these operators have been already studied in ref. [12]. These are weakly constrained operators and new RG-induced constraints can be derived only if they contribute to the running of more strongly constrained operators. In ref. [12] it has been shown that there is no such contribution and therefore we do not include these operators in our analysis. 4 By bosonic operators we denote those operators made out of boson fields. 5 The flat direction [25] between the operators O yu , O BB and O GG from the measurements of Higgs couplings to photons and gluons is lifted by considering the (still loose) upper limit on the cross section production of a Higgs boson in association with a pair of top-antitop quarks [18]. Stronger bounds on the Wilson coefficients of O BB and O GG can be obtained by imposing some theoretical priors on the value of the Wilson coefficient of O yu but we did not consider these stronger bounds here and we can safely ignore the operator O yu in our analysis. Before defining our choice for the dim-6 operator basis, let us specify the subset of independent operators on which we concentrate and which are part of the basis. This subset, which has the property that it can efficiently parametrize dim-6 contributions to the observables specified above, is given in Table 1. The basis therefore contains a total of 14 CP-even bosonic operators, notice however that O 6 does not contribute to any of the observables we are interested in, neither at tree-level nor by RG running [12]; it contributes instead to the Higgs self-coupling which however is still not directly measured. For this reason we did not include this observable in our list and did not compute its RG scaling.
The conventions in Table 1 and in the rest of the text are as follows. We define We have taken the hypercharge of the Higgs Y H = 1/2 and τ a = σ a /2 are the SU (2) L generators in the fundamental representation.
Note that the four precision parametersŜ,T , W and Y , generated in our basis by four bosonic dim-6 operators [22,26], as we show in Section 4.2, are sufficient to describe all possible dim-6 contributions to the e + e − → f + f − observables at LEP 1 and 2, only in the limit of universal new physics. To be completely general about possible new physics scenarios it would be necessary to include two more operators that contribute to the e + e − → f + f − experiment [12,18], where the former affects the SM coupling of the Z boson to the left-handed leptons, and the latter affects the measurement of G F (recall that the super-indices denote the fermion family). There are enough measurements to simultaneously constrain all six operators at the per mille level [27]. The RG contributions of {O L , O 1,2 LL } to the other operators have been already computed and can be found in ref. [12]. We have not studied possible RG-contributions of the operators of Table 1 to {O L , O 1,2 LL }, such RG-contributions could be used to impose some bounds on the weakly constrained operators of Table 1, since {O L , O 1,2 LL }, are constrained at the permil level [18]. Such an analysis would require computing many more elements of the full anomalous dimension matrix as well as enlarging the list of observables under consideration; this analysis would be interesting but beyond the scope of the present project.
The operators in Table 1 have been grouped in two different categories, corresponding to operators of the form (SM current) × (SM current) (left box) and operators which are not products of SM currents (right box). There are also 6 CP-odd counterparts of the operators in the second box which complete the list of bosonic operators of our basis, 20 in total. The current-current operators can be related to each other and to other fermionic currentcurrent operators, using the SM equations of motion (EoM) or, equivalently, by performing field redefinitions. As we discuss in the following paragraphs, this means that we have to be careful in choosing the other operators in our basis to ensure that there are no redundancies. As we discuss in Appendix B, these relationships give us an important consistency check on the anomalous dimension matrix we obtain.
Although we are interested in the anomalous dimension matrix of the 13 operators in Table 1, we have to define the complete basis of dim-6 operators that we are using. This is because, as we shall see, under RG scaling many redundant operators not in our basis, including operators containing fermions, can be generated radiatively. These operators then need to be redefined away in terms of the ones in our basis. To clearly identify these redundant operators it is thus necessary to unambiguously define our full basis including the fermionic ones. We do this in the following way: first we include the operators of Table 2 in ref. [12]. 6 Now the set of operators is an over-complete basis since it contains 20 bosonic operators + 44 operators with fermions = 64 operators in total. As shown in ref. [15], the dim-6 basis contains a total of 59 operators (for a single family), therefore there are 5 redundant operators which we can remove. Performing field redefinitions, or equivalently using the EoM's, we can trade the three four-fermions operators of the first family , O 2B } of our basis and the operators of the first family are removed in favour of the bosonic operators in Table 1, see Appendix A for more details. This completes the definition of our dim-6 operator basis, for one family. In the present work, we denote by F and f the fermion SU (2) L doublets and singlets, respectively, the subscripts R or L denote right or left-handed spinors. We put the index i (indistinguishably as a superscript or subscript) to denote either of the three SM families or, in some cases, to denote a particular SM fermion. Then, when convenient, we shall replace F by L or Q to denote an SU (2) L lepton or quark doublet, respectively, and replace f by either {e, u, d} for the right-handed fermions of the first family, and so on for the other families. To generalize the basis to three families one has to add extra four-fermion operators and take into account the different flavor indices structures. Nonetheless, these extra operators do not affect our results. Let us comment on bases of common use in the literature. The set of operators is in one-to-one correspondence with the operators used in ref. [6] {O and with the ones used in ref. [28] {O Our basis has the advantage that the anomalous dimension matrix of the sector [11]. The anomalous dimension in the other bases is given in Appendix C. As the SILH basis [28], our basis also separates the operators generated at tree-level from the ones obtained at the radiative level only, when the new physics degrees of freedom, assumed to be weakly coupled, are integrated out [11]. When the Higgs emerges as pseudo Nambu-Goldstone boson, the SILH basis further makes the distinction between a loop involving new-physics interactions and a loop involving SM interactions only.
In this paper, we are limiting ourselves to the set, B 1 , of 13 operators appearing in Table 1 (omitting O 6 that does not contribute directly to the 13 physical observables we are studying). We compute the running of B 1 into B 1 . If the remaining set of independent operators, needed to complete the basis specified above, is denoted by B 2 , there could also be i) a running of B 2 into B 1 , ii) a running of B 1 into B 2 and of course iii) a running of B 2 into itself. The first effect would reflect itself in new RG contributions to our list of low-energy observables; under our hypothesis of no-tuning (or no correlations) among the different RG contributions these effects do not change our RG-induced bounds on the operators in B 1 . In principle new RG-induced bounds on some operators in B 2 could be obtained, however we already commented on the fact that this is not the case for O 6 and O y u,d,e . The second effect could, in principle, allow us to obtain new RG-induced bound on the operators in B 1 via the mixing to some tightly constrained operators in B 2 , for example via the mixing to O L and O 12 LL , as we mentioned above. The study of these effects would be an interesting generalization of our ideas but would require the computation of the full anomalous dimension matrix and a complete phenomenological analysis of all the observables relevant to the dimension-6 operators, which is beyond the purpose of the present work.

One-loop scaling of EW and Higgs operators
In general, quantum effects mix all the operators among themselves when going from the scale of new physics down to the scale at which the experimental measurements are performed. However, the 3 operators with gluons, O GG , O 2G and O 3G , constitute a separate sector that does not mix with the other 11 bosonic operators at one-loop. 7 So, even if O GG affects Higgs physics by controlling the dominant production mode of the Higgs boson at the LHC, it can be treated separately from the 3 other Higgs observables we are interested in here. Furthermore since the Higgs self-interactions have not been measured yet, and since O 6 does not enter into the anomalous dimensions of any dim-6 operator other than itself, it can also be omitted from our analysis. For the Higgs-and EW-sector RG study, we can thus restrict to the following set of 10 dim-6 operators and compute the corresponding anomalous dimension matrix We include all the one-loop contributions proportional to c i and depending on where g , g and g s are the respective U (1) Y , SU (2) L and SU (3) c gauge couplings, λ is the Higgs quartic coupling and y t is the Yukawa coupling of the top quark, i.e. we neglect the contributions proportional to the Yukawas of the light fermions (y b /y t ∼ 0.02, y b is the bottom quark Yukawa). The couplings are normalised such that whereH = iσ 2 H * , Q L is the third family quark doublet (t L , b L ) T whose weak hypercharge is Y L = 1/6, t R is the right-handed top quark of weak hypercharge Y R = 2/3 and y l denotes the Yukawa couplings of the fermions lighter than the top. The kinetic term L Kin contains covariant derivatives, defined in the previous section, that determine the couplings of the Higgs doublet and fermions to the gauge bosons as well as gauge bosons self-interactions. We regularized the loop integrals using dimensional regularisation and used MS subtraction scheme. We performed the computation in the unbroken phase of the SM and in the background field gauge, with the gauge fixing term where δA = {δB, δW, δG} is the quantum field with respect to which the dim ≥ 4 SM action is path-integrated and D In Table 2, we give the one-loop anomalous dimensions of the operators of Eq. (3.1), in the basis defined in Section 2. 8 We have defined A common effect encountered while computing the RG scaling of the above operators is the appearance of counter-terms which correspond to dim-6 operators that are not in our basis (the computation does not know our choice of basis) [12]. These radiatively-generated redundant operators need to be redefined into operators present in our basis. Upon redefinition, these redundant operators contribute to the anomalous dimensions of the operators in our basis at the same order as other direct contributions coming from one-particle-irreducible graphs. For details on the radiatively generated operators and how we deal with the redundant ones, see Appendix A. Notice that the matrices of Table 2 already contain these indirect effects. This ensures that the result is gauge invariant and indeed we checked that the result is independent of the gauge fixing parameters ξ A of Eq. (3.4). Apart from gauge invariance, there is another non-trivial consistency check that we have performed. The current-current operators in the left box of Table 1 can be related to each other and to other current-current operators containing fermions by using the SM EoM, or equivalently by carrying out field redefinitions. In a hypothetical theory without fermions 9 , some contributions of the operators in the left box of Table 1 would vanish upon the EoM, i.e. they would form an over-complete set of operators. This would also imply relationships between independently computed entries in the anomalous dimension matrix or, in other words, the anomalous dimensions of this over-complete set is invariant under changes in the field coordinates that respect the SM gauge symmetries. Our matrix passes this consistency check as we shall discuss in detail in Appendix B. We emphasize that the set of 59 operators introduced in Section 2 is a basis, i.e. it does not contain any redundant operators; it is over-complete only in the hypothetical theory without fermions.
Some parts of the anomalous dimension matrix presented here, have been calculated in previous literature [4][5][6][7][8][9][10][11][12][13][14]. In some cases these previous computations use methods different from ours, but we find complete agreement in the final results. We present a detailed comparison with previous literature, including a discussion about the difference in our methods in Appendix C.

RG-induced contraints on EW and Higgs observables
In this section we discuss the possibility to use the RGE's to derive constraints on the Wilson coefficients at the weak scale by requiring that none of the RG contributions to these weak-scale Wilson coefficients exceeds the direct bounds [6]. Since the RGE's mix various operators, it becomes possible to put tight constraints on operators loosely constrained by direct measurements via their RG contributions to more severely constrained operators. Then, in Section 4.2, we apply our method and use EW precision data, triple gauge couplings measurements and Higgs data to derive RG-induced bounds on the set of 10 observables we are interested in. Renormalizing, order by order, the effective action, the logarithmically divergent terms computed in the previous section are absorbed in the definition of the renormalized Wilson coefficient. If one is interested in obtaining bounds on the Wilson coefficients at the low scale ∼ m H , the only effect of the 1-loop diagrams are small finite terms, proportional to ∼ log m H /m Z , which we did not compute here. Allowing for arbitrary cancellations in the definition of the renormalized coefficients renders the 1-loop effects small and the indirect bounds which can be obtained in this way are quite weak [20] and not competitive with direct bounds from Higgs physics and anomalous TGC measurements. We follow a different approach, already outlined in ref. [6]. We are interested in obtaining indirect bounds on the UV value of the Wilson coefficients from low-energy experiments, in this case the 1-loop effect is enhanced by ∼ log Λ/m H . Moreover, we assume that no tuned cancellations (or correlations) are present in the definition of the renormalized coefficients and require each log-divergent term not to exceed the direct bounds. In this way, our indirect bounds are much stronger than in ref. [20] and, more importantly, are useful in order to obtain insight into the UV physics. In fact, if any of our RG-induced bounds would be violated by a direct measurement this would imply a particular pattern of cancellation (or correlation) in the UV dynamics.

How much fine-tuning is needed to accommodate the data?
The electroweak and Higgs observables we are interested in (specified in Section 4.2) receive contributions from a particular linear combination of the dim-6 operator's Wilson coefficients, suitably multiplied by the SM couplings: where κ i is the SM contribution, the c k 's are the Wilson coefficients and ω ij is a matrix containing the SM couplings and ratios of scales (ω ∼ O(m 2 W /Λ 2 )). We definedĉ i as the linear combinations of the Wilson coefficients which contribute directly to each observable (obs) i and we shall refer to them in the following as observable couplings, with a slight abuse of language. If the new combinationsĉ i are independent, this corresponds to a change of basis such that to each operator corresponds an observable; we shall call this the observable basis.
As an example, consider the process h → γZ which receives a contribution from the SM (in this case at one loop) as well as a direct contribution from a linear combination of the dim-6 operators. We parametrize this contribution with the observable couplingĉ γZ , to be defined in Eq. (4.18), which is related to the Wilson coefficients of our basis as (c θ W and s θ W are respectively the sinus and cosinus of the weak mixing angle θ W ) The above relation defines the coefficients ω γZ,j for this particular observable. Now, suppose that this set of observables receives lower and upper bounds from experimental measurements: The observable couplingĉ i (m h ) (constrained at low energy) is related, through the running, to the high-scale value of the Wilson coefficients c j (Λ), which is not directly known since it is determined by the BSM degrees of freedom that have been integrated out. The matrix ω ij (m h ) also runs with the scale (in the example of Eq. (4.2) this would be the running of g, g and v inside m W and θ W ), however we are not interested in such a running because ω ij is determined by measurements performed at the EW scale and because, for the purpose of this work, we are not interested in the UV value of the SM couplings. This is the reason why we have not taken care of the contributions of the dim-6 operators on the SM couplings, parametrized by κ i in Eq. (4.1), which would only be necessary if we wanted to relate ω ij (m h ) to ω ij (Λ) at the order we are working. This discussion leads us to define the scale-dependent observable couplings aŝ and γ kl is the matrix computed in the previous section. Our interest in Eq. (4.5) is twofold: we want to find instances where a less constrained operator can mix with a more constrained one by appearing in its RGE's and secondly (but closely related), to learn about the new degrees of freedom at the matching scale. In the following we shall work at leading-log order, which is fine if the hierarchy between the new physics scale Λ and the EW scale is not too big.
The fundamental assumption we make in order to obtain an indirect constrain on thê c j (m h ) through the RG is that we require each term in the sum on the r.h.s. of Eq. (4.5), proportional to some coefficientĉ j , to be contained in the experimental bounds associated to the observable δ(obs) i | m h : where we defined δ i =γ ii /(16π 2 ) log(Λ/m h ) and in the last line the index is not summed over. 10 We have also used the fact that substitutingĉ j (Λ) forĉ j (m h ) in theγ ijĉj term of Eq. (4.5) amounts to corrections O (4π) −4 log 2 (Λ/m h ) that are beyond our precision (the same is true for the evaluation of γ ij ). Notice that this assumption is not only a requirement of the absence of fine-tuning but also an hypothesis on the UV physics, since particular relations, due to symmetry or dynamical accidents, between those combinations could be generically found when considering a BSM theory. From our bottom-up approach we parametrize also this absence of correlations as an absence of tuning. From Eq. (4.7) we can put bounds on the matching-scale Wilson coefficients c j (Λ): notice that, as expected, they grow quadratically weaker with the increase of the UV scale Λ since ω −1 ∼ Λ 2 /m 2 W . Using Eq. (4.8), instead, we can put an RG-induced bound on the observable δ(obs) j | m h using the direct constraints on δ(obs) i | m h , Eq. (4.3): (4.10) The indirect bounds in Eq. (4.10), grow logarithmically stronger with the increase of the UV scale Λ. However, since the expected effects from new physics decrease quadratically with Λ, assuming order one coefficients c i , even if the RG-induced bounds on the observables become slightly stronger, their power in investigating the UV degrees of freedom becomes much weaker for higher values of Λ, as is clear from Eq. (4.9). It might seem that these bounds are not significant because of the loop factor in the above equation; all the i 's are, however, not of the same order and if | low,up i | | low,up j |, the bound in the above equation can be stronger than the direct bound on δ(obs) j | m h , in spite of the loop factor. The RG-induced bounds are, thus, significant only when a weakly constrained coupling appears in the RGE of a strongly coupled one.
Once new physics effects will be, hopefully, observed and the constraints of Eq. (4.3) will not include the zero value in the allowed interval (0 < low , another interesting information that could be extracted from RG effects is a quantification of how much tuned, among themselves, are the electroweak and Higgs observables. First of all, let us define the fine-tuning in an observable as [29] 10 In the following we shall denote with a hat all repeated indices which are not summed over. where in the second step we separated the diagonal contribution from the off-diagonal ones and, for the diagonal term, we neglected the loop contribution sinceĉ i (Λ) enters already at tree level and this would be its leading contribution to the tuning. In particular, the fine-tuning ∆ i will satisfy, and one might be able to conclude that a certain degree of fine-tuning among the contributions to the RG flow of some operator is necessary.

EW and Higgs observables
Let us now apply the general formulas of the previous section to the electroweak and Higgs observables we want to constrain. In Section 2 we have considered 10 EW and Higgs operators to parametrize BSM corrections to the SM Lagrangian. Let us now describe in detail the set of pseudo-observables, briefly mentioned in Section 2, that constrain all these operators and form our observable basis. These include the four electroweak oblique parametersŜ,T , Y and W , constrained by LEP 1 and LEP 2, the three anomalous triple gauge coupling (TGC) and three observables related to Higgs physics: the decays to γγ, γZ and a universal rescaling of all the branching ratios [18]. To derive the RG-induced constraints on these observables we first need to relate them to the operators in Eq. (4.13), that is define the transformation matrix, ω ij , from the basis in Eq. (4.13) and to the observable basis. We begin with the electroweak precision observables constrained by measurements at LEP1, LEP2 and Tevatron. The first step of the analysis is to fix the SM parameters g, g and v by the three most precise measurements: the Fermi constant G F in muon decays, the fine-structure constant α em and the Z-boson mass m Z . With the input parameters fixed, the SM gives predictions for observables such as Z-pole measurements at LEP 1, the Tevatron measurement of the W -mass and LEP 2 measurements of the e + e − → f + f − cross-sections. New physics can affect this analysis by either changing the relationship between the input parameters g, g and v to the measurement of G F , α em and m Z or by directly contributing to the other measurements. All the deviation in the above observables induced by the operators we consider, Eq. (4.13), can be parametrized by theŜ,T , W and Y parameters [22] The contribution of the Wilson coefficients of the operator set in Eq. (4.13) to the above observables is given by, The above parameters have been measured very precisely and are constrained at the per mille level. We present the 95 % CL bounds on these parameters in Table 3.
A second set of independent measurements that constrain the operator set in Eq. (4.13) are the TGC that were measured in the e + e − → W + W − process at LEP2. The phenomenological Lagrangian to describe deviations in the TGC observables, from their SM values, is 11 has field-strengthẐ µν and we use s θ W ≡ sin θ W = g / g 2 + g 2 , c θ W ≡ cos θ W = g/ g 2 + g 2 and e = gs θ W . Note that the above Lagrangian has only three independent parameters at the dim-6 level taken to be g Z 1 , κ γ and λ γ here; the other two can be expressed as : λ Z = λ γ and κ Z = g Z 1 − t 2 θ W κ γ . These relations are a consequence of the accidental custodial symmetry that is preserved by the dim-6 operators entering in the TGC [30]. The SM contribution is given by (g Z 1 ) SM = (κ γ ) SM = 1 and (λ Z ) SM = 0. The corrections induced by the dim-6 operators in our basis are given by: where δg Z 1 = g Z 1 − (g Z 1 ) SM and δκ γ = κ γ − (κ γ ) SM . The constraints on these TGC observables are at the percent level (see Table 3) and thus at least an order of magnitude weaker than the constraints on the electroweak parameters in Eq. (4.15). Note that, for this reason, in Eq. (4.17) we have ignored contributions to the e + e − → W + W − process from the couplings in Eq. (4.14).
Higgs physics provides the three remaining observables for our observable basis. We consider the branching ratios h → γγ/Zγ and the correction to the Higgs kinetic term, (4.18) 11 Note that in the previous version of the paper the deformations related to δg 1 Z and δκ γ were defined with a sign opposite to that used in the literature. We have changed this, and now we use the conventional definitions. The above coefficients, in terms of the dim-6 operator's Wilson coefficients are given bŷ We present the constraints on these three observables in Table 3. The couplingĉ γγ is constrained at the per mille level although the constraint on the SM diphoton width has been measured only with O(1) precision. This is because the SM width is already one-loop suppressed and thus the current O(1) precision of measurement corresponds toĉ γγ ≈ 10 −3 . The correction to the Higgs kinetic termĉ H on the other hand is poorly constrained. This is becauseĉ H causes a universal shift in all the Higgs couplings and thus drops out from the branching ratios. Moreover, if only gluon fusion production channels are considered, the coupling c GG mimics the effect ofĉ H . Therefore, to disentangle the effect of c GG and constrain c H , Higgs production cross-sections in different channels have to be compared; in particular the weakly sensitive vector-boson fusion (VBF) channels have to be considered.

Coupling Direct Constraint
Based on their precision of measurement, the observables can be divided into at least two groups. In the first group, containing highly constrained operators, we have the four electroweak parameters and the Higgs diphoton coupling (see Table 3), 20) which have been measured at the per mille level. In the second group we have the hγZ coupling, the couplings related to the three TGC observables κ γ , g 1 Z , λ γ andĉ H , . We present the anomalous dimension matrix in the observable basis in Table 4. Using this, and fixing Λ = 2 TeV, we write numerically Eq. (4.5) as We can now derive the RG-induced constraints by using Eq. (4.10) assuming no fine-tuning among the different terms in the RGE's.
The strongest RG-induced constraints come from the direct bounds on theŜ,T , W and Y parameters, i.e. the first four lines in Eq. (4.22). We require that each observable coupling individually satisfies the four RG-induced constraints from these electroweak precision parameters simultaneously. It is very important to take into account the experimental correlations betweenŜ,T , W and Y while imposing these bounds [33][34][35]. Note that the RG-mixing contributions toĉ W andĉ Y , from the couplings in the weakly constrained group in Eq. (4.21), is either absent or accidentally much smaller than the ones toĉ S andĉ T (see the RG contributions toĉ W andĉ Y in the third and fourth row of Eq. (4.22)). We, therefore, look at the constraints on theŜ −T plane taking W = Y = 0. We use theŜ −T ellipse in ref. [31], which assumes W = Y = U = 0, to derive our constraints. We present these RG-induced bounds and compare them with the direct bounds in Table 3 and in Figure 1. We find that for each of the couplings in the second group we can derive a RG-induced constraint stronger than,  Table 3; the line is green (red) for positive (negative) values of the parameters.
or of the same order of, the direct tree-level constraint. We also obtain RG-induced bounds from the direct constraint onĉ γγ using the fifth line in Eq. but at present these bounds are weaker than those from the direct bounds on electroweak parameters. Let us briefly comment on alternate choices for our observable basis. In general, a change of observable basis modifies the anomalous dimension matrix of Table 4, also for the observables which were maintained in the basis. Thus, the RG-induced constraints we have derived, are applicable only to our particular choice of observables, and for an alternate choice the analysis must be repeated. 12 For instance, the Higgs decay observables related to h → W + W − , ZZ decays could have been alternatively chosen as part of our observable basis instead of two of the TGC observables (κ γ and g Z ) but we have kept the TGC in our basis as they are measured 12 Note that for our choice of observable basis, h → γγ does not receive a contribution from theŜ parameter even though there is a dependance on c W B in the anomalous dimension but c W B is actually reconstructing the δκ γ parameter. more precisely than these Higgs decay observables. This situation is likely to continue in the future. Although, observables like the relative deviation of h → W + W − , ZZ with respect to the SM would be strongly constrained at the 5 %(3 %) level at the LHC with 300 fb −1 ( 3000 fb −1 ) data [36], the bounds on TGC are also expected to become stronger by an order of magnitude at the LHC [36] so that the TGC would still be more precisely measured than these Higgs observables. At linear colliders the Higgs h → W + W − , ZZ is expected to be measured at the level of 0.5 % [36] and the TGC observables at the 10 −4 level [37]; again the TGC observables would be more constrained.
Finally, let us discuss the future prospects for these RG-induced effects. In the future, as the measurement of the observables we have considered becomes more and more precise, it may be possible to detect signs of new physics. In this case, since some of the observables in Table 3 will be non-zero one would expect a deviation, via RG-mixing, also in other observables, unrelated at tree level. Note that according to future projections,ĉ γγ , the TGC observables (ĉ κγ ,ĉ gz ) andĉ γZ would be measured at the 10 −4 level [36,37] at linear colliders and thus all these observables would be sensitive to RG-induced mixing effects of the couplings in Eq. (4.21), if they are above a minimal value. 13 We present these minimum values in Table 5. If, instead, a deviation is detected in some observable but no such RG-induced deviation in other observables is detected at the level hinted by our analysis, then this would indicate a tuning (or a correlation) among the various RG contributions to the direct measurement, see Eq. (4.11). Take, for example, the first row of Table 5. Suppose we measure the deviation c λγ ∼ 1 × 10 −2 , a value larger than the minimum value presented in Table 5, while instead h → γγ would still remain compatible with zero with the reported sensitivity. From Eq. (4.11) we would than conclude that a fine-tuning of the order ∆ γγ 5 would be necessary to accommodate the data, or that some particular correlation in the UV physics is needed to induce such cancellation.

Scaling of the gluon operators
In this section we shall extend the results of the previous sections and present also the scaling of the bosonic operators that contain gluons, as defined in Table 1: The anomalous dimension matrix is shown in Table 6, where the c 3G self-renormalization has been taken from refs. [4,5]. This matrix already contains the effect of the redundant operators 13 Future prospects for measurements at the Z-pole predict an enhancement of the precision, with respect to the present one, of about one order of magnitude for ILC [37] and two orders of magnitude for TLEP [38], depending on the observable. Moreover, from runs at energy √ s ∼ 2m W , the measurement of the W mass is predicted to became more precise by one (ILC) or two (TLEP) orders of magnitude. This will imply an enhancement of the precision in the oblique parametersŜ,T , W and Y . A more detailed study of these future prospects is beyond the scope of this paper, since our aim is only to show some examples for future applications of the general idea of RG-induced bounds. Table 4: Anomalous dimension matrix in the observables basis. We defined t θ W = tan θ W .

Direct
Future that are generated radiatively and, upon eliminating them, modify the RG of the operators in Table 1, see Appendix A for details.
In the same spirit of Section 4, let us now turn to the observables which are sensitive to these operators and review the present constraints. The Wilson coefficient c 2G can be put in one-to-one relation to the parameter Z introduced in ref. [22] (analogous to the W and Y electroweak parameters): A bound on this parameter has been obtained by an analysis of dijets events at LHC [39]: A bound on c GG can be obtained from the analysis of the Higgs production cross section at LHC. The relevant phenomenological Lagrangian is The most recent bound, obtained in ref. [18] after marginalizing over the other deviations from the SM, readsĉ The coefficient c 3G , analogous to the SU (2) L counterpart c 3W , would contribute to the anomalous triple gluon couplings. These effects can be measured at LEP, Tevatron and LHC, for example via top-quark pair production, see for example ref. [40] where it is estimated that LHC should be able to put a bound |ĉ 3G | ≡ |c 3G | m 2 W /Λ 2 0.1. As can be seen in Table 6, no mixing to (or from) these gluon operators is present among the operators we considered in Table 1, the only exception being a contribution from c 2B to c 2G which, however, is not very interesting since c 2B is already very well directly constrained by the oblique Y parameter. For this reason, we are not able to cast any indirect constraint using these gluon operators.

Conclusions
We computed the scaling and mixing of 13 dim-6 deformations of the SM affecting EW precision observables (4), anomalous EW triple gauge boson couplings (3), QCD observables (2) and Higgs decays (4). This computation has important phenomenological implications. Particularly interesting is the RG-mixing induced among 10 of these observables (the 2 two QCD observables and one Higgs observable, namely Γ(h → gg), constitute a separate sector that does not mix in a relevant way with the severely constrained EW observables.). These 10 different observables are constrained at very different levels of precision. For example, whereas the electroweak precision observables and the operator coefficient related to the h → γγ partial width are constrained at the per mille level, the TGC and the 2 other Higgs observables are constrained at the percent level at most. As we run down from the new physics scale to the lower scale of experiments, quantum effects mix the observables and the most severely constrained ones receive a contribution from the ones allowed to deviate the most from the SM predictions. These RG-contributions could in principle be of the same size or even larger than the direct experimental bounds, in other words, the difference in the experimental sensitivities can compensate for the RG-loop factor. Requiring that these RG-contributions do obey individually the direct bounds, i.e. dismissing any possible tuning/correlation among the various RG-terms, we can derive some indirect RG-induced bounds on the weakly constrained observables from the direct measurement of the severely constrained ones. This analysis is particularly relevant for the TGC and the universal shift of the Higgs couplings, as reported in Table 3.
We also looked at the future prospects of these RG-induced effects. If a deviation from the SM is observed in some of the observables we considered, in the absence of tuning one would expect a deviation, due to these RG effects, to appear also in other seemingly unrelated observables. If, instead, these RG-induced deviations are not observed, it would mean that some tuning is needed, or it would indicate some correlation among the higher dimensional operators pointing towards a particular structure of the new physics that has been integrated out. We have presented the projected future experimental sensitivity to these RG effects in Table 5.
The first run of the LHC ended beautifully with the discovery of the Higgs boson and initiated an era of measurements in the EWSB sector that remained only indirectly constrained for several decades. With the next run of the LHC and the high-luminosity program will start an era of precision that will lead certainly to a better understanding of the Higgs sector itself and also, hopefully, to the first glimpse of the new physics laying beyond the Standard Model. We hope that the results we presented in this paper will be a powerful tool in that quest.
Note added: While this paper was being submitted, the work [41] appeared. It computed the gauge-coupling dependence of the anomalous dimensions among the dim-6 operators.

A Dealing with redundant operators
In this appendix we explain in detail the anomalous dimension matrix presented in the main body of the paper, Tables 2 and 6. As remarked in Section 2, a common effect encountered in the computation of the scaling of the dim-6 operators is the appearance of counter-terms that correspond to operators not included in our basis, i.e. operators that are redundant for the description of physical processes. In particular, the set of 13 operators we are interested in, not only mix among themselves under the RG flow but also generate redundant operators that are not included in our basis (defined in Section 2). In this appendix we first give a pedagogic example of radiatively generated redundant operators, Section A.

A.1 Example of radiatively generated redundant operators
As a first step, let us give a detailed example of the generation of redundant operators by the ones in Eq. (A.1). Consider the renormalization of the vertexē R − e R − B ν by the operator O 2B . There is only one possible diagram, depicted in Fig. 2(b), which can give contributions to any of the operators or to the three-point vertex of the operator It can be easily checked that there is no other operator with the same field content which is also independent from the ones in Eq. (A.2) and Eq. (A.3). As for a CP-odd version of O e BR is of not concern to us since it is clear that the diagrams we are considering cannot violate CP. The crucial point of this discussion is that the above operators are not contained in our basis, therefore one has to redefine them back to the ones in our basis, giving a contribution in the anomalous dimensions. These indirect contributions of O 2B to the anomalous dimensions of the bosonic operators are of the same order as the direct contributions computed via oneparticle-irreducible diagrams, it is therefore necessary to keep track of all such effects in order to have a consistent calculation.
The computation of diagram (b) in Fig. 2 using dimensional regularization with D = 4 − 2 . O i represents the Feynman rule of the operator O i with the external states of diagram (b). We obtain Diagram (a) in Fig. 2 gives univocally the contribution of O 2B to O e K3R , since this is the only dim-6 operator with only {ē R , e R } as external legs. Since we are working in the background field gauge, we expect that all vertices of a gauge invariant operator should be renormalized in the same way. 14 For this reason, and from the previous computation in Eq. (A.5), we already know what that the result of diagram (a) should be: γ In the following section we shall study how these redundant operators, generated by the RG flow, can be redefined into the ones of our basis. For instance, we shall see that the contribution from O 2B to O e BR described above has to be included as a contribution to the running of O B and O 2B , Eq. (A.18).

A.2 Anomalous dimension matrix
The relevant redundant operators that are radiatively generated by those in Eq. (A.1) are: By relevant we mean those radiatively generated redundant operators that modify the Wilson coefficient of the operators in Eq. (A.1) when the former operators are redefined into operators in our basis, defined in Section 2.   Tables 7, 8 and 9 below arise from one-particle-irreducible Feynman diagrams, i.e. it is the one-loop renormalization of the Effective Action.
In Table 7 and in the background field gauge. N G = 3 is the number of generations. The contributions not shown are either zero or proportional to the Yukawa coupling y l of any fermion lighter than the top. Notice that in Table 7 we have gone beyond the strictly necessary computations to obtain the anomalous dimension matrix and also included the contributions of the operator O r , that is redundant with respect to our basis; their contributions are used for a crosscheck in Appendix B.
In Table 8   ref. [4]. Their contribution to the running of the redundant operators in Eq. (A.6) that we have not written are either zero or proportional to y l . Lastly, in Table 9 is ξ-dependent. This should not come as a surprise, even if we work in the background field gauge, where the counter-terms are gauge invariant. The reason is that at this point of the computation we still have redundant operators generated by the flow. By definition, in an over-complete basis that contains redundant operators only certain combinations of the Wilson coefficients enter in the physical observables. Hence, it is only after these physical combinations of the Wilson coefficients are taken, that the computation is guaranteed to be and should be gauge invariant. For instance, in Section A.3 we show that upon redefining the redundant operators in terms of operators in our basis the ξ dependence of the anomalous dimension vanishes. This subtlety is well known and, for instance, it also appears in the context of Non-Relativistic QCD, where the running of the Wilson coefficients is gauge independent only when the redundancy of different operators is taken into account [42]. This has also been recently stressed again in ref. [13]. γc H 3 4 g 2 g 2 + 4g 2 3 4 g 2 3g 2 + 4g 2 − 6λg 2 − 3 8 g 2 g 2 + 4g 2 − 3 8 g 2 g 2 (3 + 2ξ W ) + 4g 2 + 3λg 2 γc T − 9 4 g 2 g 2 − 6λg 2 − 9 4 g 2 g 2 9 8 g 2 g 2 + 3λg 2  Table 9  Table 9 γ c u,d K3R − 4 3 g 2 s 0 0 Table 9 0 Table 9  Table 9 γ c Q BL − 10 9 g 2 s Y Q 0 0 Table 9  Table 9 γ c u,d BR − 10 9 g 2 s Y u,d 0 0 Table 9 0 Table 9  Table 9 γ c Q BL − 4 3 g 2 s Y Q 0 0 Table 9  Table 9 γ c u,d BR − 4 3 g 2 s Y u,d 0 0    Table 6).

A.3 Removal of the radiatively-generated redundant operators
We now turn in to discuss how to deal with each operator in Eq. (A.6) and their effect on the operators of Eq. (A.1).
The easiest way to deal with the redundant operator [15] is by means of the identity 15 γ µ γ ν γ ρ = g µν γ ρ + g νρ γ µ − g µρ γ ν + i µνρσ γ σ γ 5 ; (A.9) one finds Then, using the fermion's EoM which is a dipole operator, where σ µν ≡ i 2 [γ µ , γ ν ]; using again Eq. (A.9) in the second term of the right hand side of Eq. (A.10) Therefore, Eqs. (A.10)-(A.12) and analogous manipulations, are equivalent to the following shifts (c i → c i + δc i ) in the following Wilson coefficients: The Wilson coefficient of the dipole operators are also shifted, see Eq. (A.11), however, we can not conclude that the dipoles are renormalized by the set of bosonic operators we considered because we did not compute direct contributions, those coming from one-particle-irreducible diagrams.
Then, for the operator O f i K3R , consider the field redefinition (A.14) 15 We use the conventions of Peskin & Schroeder textbook.
where X µν = g Y F i B µν + gW a µν τ a + g s G A µν T a , being τ a and T A the SU (2) L and SU (3) c generators in the fundamental representation, respectively. Then, by inserting the fermion's EoM in the first operator in the right hand side of Eq. (A.14) one gets operators of the type L Yuk |H| 2 and the operator y we do not care about the latter (proportional to y f i ) since our basis choice of Section 2 was to remove the operator O f i R corresponding to a light fermion. Performing an analogous analysis for O F i K3L we reach the same conclusion: neither of the two operator's scaling affects the anomalous dimension of the set of bosonic operators in Eq. (A.1). As in the case of O W L,BL,BR , the same comment applies here: eventhough the Wilson coefficient of the dipoles is shifted by the above manipulations, we do not conclude that they are renormalized by the bosonic operators. Now, the remaining operators (corresponding to the third, forth and fifth line of Eq. (A.6)) are redefined into our basis by performing field redefinitions. Consider the 37 independent field redefinitions We use the freedom given by the field redefinitions to set to zero the following 37 coefficients: c r , c K4 , c and Eq. (A.13) has already been taken into account. This completes the removal of the operators in Eq. (A.6) in terms of the bosonic operators.
As we have just shown, upon eliminating the redundant operators the Wilson coefficients of the operators of Eq. (A.1) are shifted in such a way that the anomalous dimensions are redefined as after setting c 0,i red = 0, being c 0,i red the tree-level Wilson coefficient of any of the operators in Eq. (A.6). The anomalous dimensions of the remaining bosonic operators, that are not of the form (SM current)×(SM current), are not redefined. In this way we can go back to our original basis taking into account that some operators are generated radiatively even if we set their Wilson coefficient to zero at the matching scale. In the main body of the paper, Tables 2 and 6, we gave the physical anomalous dimensions obtained using the right hand side of Eq. (A.20). As announced in Section A.2, the ξ dependence cancels out in the physical combinations of γ c i 's, which can be easily checked using Eq. (A.20).

B Field Reparametrization-Invariance Crosscheck
There is a useful consistency check that can be done to the results presented in Tables C 6 ≡ c 6 + 2c r + g 2 (2c W − c 2W ) + 4λc K4 .
Physical observables depend on shift invariant combinations of couplings, which we denote by a capital C i . Also, a key property is that the anomalous dimension of a shift invariant combination of couplings is a function of shift invariant combinations of couplings only 17 This is precisely the cross-check that can be done to the results computed in Tables 7 and 9. And indeed it is easy to check that: as it should, given the fact that O 6 does not renormalize O r , O H , O T . As its clear from the discussion above, to compute Eq. (B.5) one has to insert the Higgs and gauge bosons anomalous dimensions and the gauge beta functions without the contributions of the fermions: in the background field gauge and the superscript nf stands for no fermions, to distinguish them from their SM counterparts. Notice also that in Eq. (B.5) the ξ dependence exactly cancels, as it should, rendering the result independent of the gauge fixing term of Eq. (3.4).

C Comparison with previous literature
Let us now put into context the results for the anomalous dimensions presented in this paper. The first paper in the literature with a similar spirit to ours is ref. [6], followed by ref. [8] and more recently by refs. [20] and [21] Table 11; they correspond to the contributions of {c HB , c HW , c BB , c 3W , c W W } to {γ c 2B , γ c 2W , γ c W B , γ c T }, we find complete agreement. We want to stress here that the approach we followed to compute the running of the Wilson coefficients is somewhat different than in ref. [6]. We computed the effective action (only oneparticle irreducible graphs) in the background field gauge starting from a basis of operators; then, along the RG flow operators not included in the basis (like O L ) are generated. These 17 See ref. [11] for a more detailed discussion. 18 Notice that we have chosen different normalizations for the operators, different conventions for the covariant derivatives and different names for the operators with respect to ref. [6]. 19 In are redefined into our basis, and they are interpreted as the vertex (e L σ a γ µ e L W a µ ) corrections that are computed in refs. [6,7] to make the result physical and hence gauge invariant. We believe that the approach we followed to compute the anomalous dimensions is somewhat more systematic when dealing with the renormalization of other operators than the oblique ones.
Contributions from O H to the T and S-parameters are given in ref. [9]. Then, the separate contributions of {O T , O H } to {O B , O W } are given in ref. [11]. The 3 × 3 matrix of anomalous dimensions for the operators {O BB , O W B , O W W } has been computed in ref. [10], together with its CP-odd counterparts. Then, in ref. [11] it is shown that the 3 × 3 anomalous dimensions matrix computed in ref. [10]  In ref. [12] several anomalous dimensions were computed, some of them overlap with the work presented here. These are the contributions of {O H , O r } to the anomalous dimension of any dim-6 operator. The contributions from operators containing fermions to the anomalous dimensions of any interesting operator for Higgs physics or EW precision tests are also computed in ref. [12].
The Yukawa dependence of the anomalous dimensions matrix of the dim-6 SM operators is given in ref. [14]. However, notice the Yukawa dependences needed in the present paper to derive RG-induced constraints come only from the wave functions of the SM particle's field or are proportional to small Yukawas. Tables 11 and 12 show the results for the anomalous dimensions matrix presented in the main body of the paper in two of the most used bases in the literature, refs. [6] and [28]. A good property of our basis with respect to the ones in the literature is that the one-loop anomalous dimension matrix is simpler, since its has a block diagonal structure.
c H c H