A bottom-up approach within the electroweak effective theory: constraining heavy resonances

The LHC has confirmed the existence of a mass gap between the known particles and possible new states. Effective field theory is then the appropriate tool to search for low-energy signals of physics beyond the Standard Model. We adopt the general formalism of the electroweak effective theory, with a non-linear realization of the electroweak symmetry breaking, where the Higgs is a singlet with independent couplings. At higher energies we consider a generic resonance Lagrangian which follows the above-mentioned non-linear realization and couples the light particles to bosonic heavy resonances with $J^P=0^\pm$ and $J^P=1^\pm$. Integrating out the resonances and assuming a proper short-distance behavior, it is possible to determine or to constrain most of the bosonic low-energy constants in terms of resonance masses. Therefore, the current experimental bounds on these bosonic low-energy constants allow us to constrain the resonance masses above the TeV scale, by following a typical bottom-up approach, i.e., the fit of the low-energy constants to precise experimental data enables us to learn about the high-energy scales, the underlying theory behind the Standard Model.

The LHC has confirmed the existence of a mass gap between the known particles and possible new states. Effective field theory is then the appropriate tool to search for low-energy signals of physics beyond the Standard Model. We adopt the general formalism of the electroweak effective theory, with a non-linear realization of the electroweak symmetry breaking, where the Higgs is a singlet with independent couplings. At higher energies we consider a generic resonance Lagrangian which follows the above-mentioned non-linear realization and couples the light particles to bosonic heavy resonances with J P = 0 ± and J P = 1 ± . Integrating out the resonances and assuming a proper short-distance behavior, it is possible to determine or to constrain most of the bosonic low-energy constants in terms of resonance masses. Therefore, the current experimental bounds on these bosonic low-energy constants allow us to constrain the resonance masses above the TeV scale, by following a typical bottom-up approach, i.e., the fit of the low-energy constants to precise experimental data enables us to learn about the high-energy scales, the underlying theory behind the Standard Model.

I. INTRODUCTION
The LHC has confirmed the success of the Standard Model (SM) with the discovery of a Higgs-like 1 particle [1], with couplings compatible with the SM expectations, and the non-observation of new states, which establishes the existence of a mass gap between the SM and possible new physics (NP) fields. This gap justifies the use of effective field theories to analyze the data, and the lack of information about the hypothetical underlying theory behind the SM invites us to follow a bottomup approach, that is, to search for fingerprints of heavy scales at low energies in a systematic way.
In this bottom-up approach the low-energy constants (LECs), or Wilson coefficients, are free parameters which encode the information about the heavy scales, whereas the construction of the effective Lagrangian (the local operators) depends on the light-particle content, the symmetries and the power counting. There is no doubt about the particle content in this case, the SM states, but there are two different ways of introducing the Higgs field, and this has consequences in the symmetries and in the power counting to be used [2,3]. One can consider the more common linear realization of the electroweak symmetry breaking (EWSB), assuming the Higgs to be part of a doublet together with the three electroweak (EW) Goldstones, as in the SM, or the more general non-linear realization, without assuming any specific relation between the Higgs and the Goldstone fields. The first option is known as the SM effective field theory (SMEFT) and it is organized as an expansion in canonical dimensions, being its leading-order (LO) approximation the dimensionfour SM Lagrangian. We follow here the second option, the EW effective theory (EWET), also known as Higgs effective field theory (HEFT) or EW chiral Lagrangian (EWChL), where an expansion in generalized momenta is followed. The LO description is given in this case by the Higgsless SM Lagrangian plus the O(p 2 ) operators that introduce the Higgs and the EW Goldstones. Note that the SMEFT is a particular case of the more general EWET framework.
At higher energies we introduce massive resonance states by using a phenomenological Lagrangian respecting the non-linear realization of the EWSB, i.e., respecting the symmetries and following the chiral expansion of the EWET. Therefore, we consider two effective Lagrangians with different particle contents: the EWET at low energies, with only the SM particles, and the EW resonance theory at high energies, with the SM particles plus heavy resonances. Both Lagrangians can be matched in a common validity region by integrating out the resonances; in other words, the EWET LECs can be determined in terms of resonance parameters. In order to obtain interesting constraints (relevant from a phenomenological point of view), it is very convenient to assume a given short-distance behavior of the unknown underlying theory. This allows us to get determinations or bounds in terms of only resonance masses.
The main purpose of this work is to combine the current experimental bounds on the bosonic EWET LECs with their determinations or limits in terms of resonance masses, in order to constrain the scale of new physics. We update the results already presented in Ref. [4], extending the resonance Lagrangian (considering P -even and also P -odd operators, bosonic and also fermionic resonances, color singlets and also color octets), softening the high-energy constraints and including the current experimental bounds. Moreover, since precise experimental measurements of the hW W coupling κ W are now available, we no-longer consider this coupling as a non-free parameter. 2 The theoretical framework is briefly presented in Section II: the Lagrangians of the EWET and the resonance electroweak theory are given in Sections II A and II B, respectively, whereas the assumed short-distance behavior is described in Section II C. Some technical details are relegated to the Appendix. In Section III the current experimental bounds on the bosonic EWET LECs are summarized. The connection between the theoretical predictions of Section II and the experimental bounds of Section III is done in Section IV, being the plots in Figures 1 and 2 the main results of our analysis. Some concluding remarks are finally given in Section V.

A. Low energies: EWET Lagrangian
The EWET Lagrangian can be organized as an expansion in powers of generalized momenta [3,[8][9][10][11]: Note that, as it has been stressed previously, the operators are not ordered according to their canonical dimensions and one must use instead the chiral dimensiond, which reflects their infrared behavior at low momenta [11]. Consequently, loops are renormalized order by order in this expansion. We collect in the Appendix the definitions of the building blocks, used to construct operators invariant under the electroweak symmetry group G, and the powercounting rules that determine their chiral dimensions. The relevant bosonic part of the LO EWET Lagrangian is given by 3 where h denotes the Higgs field, the u µ tensor contains the EW Goldstones and · · · 2 indicates an SU (2) 2 In Ref. [4], κ W was determined theoretically with the second Weinberg Sum Rule [5] of the W 3 B correlator at next-to-leading order (NLO): . This constraint was a consequence of having considered only the two-Goldstone (ϕϕ) and the Higgs-Goldstone (hϕ) cuts in the NLO determination of the oblique parameters S and T [7]. 3 An alternative notation a = κ W , b = c 2V is used in some works.
trace. Thus, κ W parametrizes the hW W coupling in SM units. Assuming invariance under CP transformations, the bosonic NLO EWET Lagrangian reads [8,9] Tables I and II display the explicit list of operators O i (P -even) and O i (P -odd), respectively.
Since the Higgs is a singlet under G, the operators of (3) can be multiplied by arbitrary analytical functions of h/v [14], i.e., their LECs are actually Higgsdependent functions that can be Taylor-expanded in powers of h/v. Given the current experimental status, from now on we consider only the first term in this expansion: F i ≡ F i (h = 0) and F i ≡ F i (h = 0). The couplings F 2 , F 11 , F 12 and F 2 are not measurable, because their corresponding operators reduce to terms already present in the electroweak and strong Yang-Mills Lagrangians. Therefore, they can be re-absorbed through a renormalization of the gauge couplings g, g and g s . Within the SM, κ W = c 2V = 1 and F i =2,11,12 = F i =2 = 0.
The operator O 1 contributes to the W ± and Z selfenergies and can then be accessed through the measurement of the so-called oblique parameters [7]. The trilinear and quartic gauge couplings are sensitive to O 1,3 and O 1,3−5 , respectively, while O 1 can modify both. O 10 generates custodial-breaking corrections to all Goldstone Green functions. The remaining operators, O 6−9 and O 3 , contribute to Goldstone vertices involving Higgs bosons.

B. High energies: resonance electroweak theory
Although the EWET power counting is not directly applicable to the resonance theory, one can construct the Lagrangian in a consistent phenomenological way,à la Weinberg [11], which interpolates between the low-energy and the high-energy regimes: the appropriate low-energy predictions are generated and a given short-distance behavior is imposed [15].
Taking into account that here we are interested in the resonance contributions to the bosonic O(p 4 ) EWET LECs, we only need to consider O(p 2 ) operators with up to one bosonic resonance field [8,9]. Moreover, we can drop the couplings to fermionic resonance fields because their tree-level exchanges do not contribute to the bosonic LECs [8].
Following the notation of Ref. [8], the dimension of the resonance representation is indicated with upper and lower indices in the scheme R  [8,9].  [8,9].
The spin-1 resonances V and A can be described with either a four-vector Proca fieldR µ or with an antisymmetric tensor R µν . Here we keep both formalisms because, at it was demonstrated in Ref. [9], the sum of tree-level resonance-exchange contributions from the O(p 2 ) resonance Lagrangian with Proca and antisymmetric spin-1 resonances gives the complete (non-redundant and correct) set of predictions for the O(p 4 ) EWET LECs, without any additional contributions from local operators without resonance fields.
The relevant resonance Lagrangian takes the form [8]: We only display the terms which contribute to the bosonic LECs we are interested in. The first line shows the interactions without resonances, those with a heavy scalar or pseudo-scalar boson, and the heavy vector and axial-vector operators with Proca fields; whereas in the second and third lines the heavy vector and axial-vector contributions with antisymmetric resonances are given, respectively.
Integrating out the heavy resonance fields in Eq. (5), one recovers the EWET Lagrangian (3) with explicit values of the LECs in terms of resonance parameters. These determinations are also shown in Tables I and II [8,9]. Note that F 4 and F 4 + F 5 are predicted to be positive, in agreement with the known requirements from causality, crossing and analyticity [16].

C. Short-distance constraints
As it has been spotlighted previously, assuming a good short-distance behavior is important because of two reasons: • From a theoretical point of view, the resonance theory tries to interpolate between the low-energy and the high-energy regimes: by construction the low-energy effective Lagrangian (the EWET in our case) is recovered when the resonances are integrated out, as it has been shown for the bosonic operators in the previous subsection; however, and in order to deal with a good interpolation with the underlying dynamical theory at high energies, shortdistance constraints are needed to implement the assumed high-energy limits.
• Moreover, and from a practical point of view, these constraints are very convenient to reduce the number of resonance parameters. In other words, without short-distance constraints Tables I and II show the determination of the fifteen EWET LECs of (3) in terms of the sixteen resonance couplings of (5) and the four resonance masses, so the interest of well-motivated constraints is evident.
The following high-energy constraints have been considered [9]: 1. Well-behaved form factors. The two-Goldstone and Higgs-Goldstone matrix elements of the vector and axial currents can be characterized by the corresponding vector and axial form factors. Assuming that they vanish at high energies, four constraints are found: (a) Vector form factor to two EW Goldstones (ϕϕ): (b) Axial form factor to two EW Goldstones (ϕϕ): (c) Axial form factor to a Higgs and one EW Goldstone (hϕ): (d) Vector form factor to a Higgs and one EW Goldstone (hϕ): 2. Weinberg Sum Rules (WSRs). The W 3 B correlator is an order parameter of the EWSB. In asymptotically free gauge theories it vanishes at short distances as 1/s 3 [17], implying two superconvergent sum rules, the so-called 1st and 2nd WSRs [5]: (a) 1st WSR (vanishing of the 1/s term): (b) 2nd WSR (vanishing of the 1/s 2 term): While the 1st WSR is expected to be also fulfilled in gauge theories with nontrivial ultraviolet (UV) fixed points, the validity of the 2nd WSR depends on the particular type of UV theory considered [18].
Resonance-exchange contributions to the P -even bosonic O(p 4 ) LECs, considering only P -even operators and the short distance constraints. Entries marked with · · · indicate that the result is the same as in Table I, without further simplification.
Using these constraints, we can sharpen the determinations of the EWET LECs in Tables I and II, writing them in terms of a smaller number of resonance parameters. Combining the two WSRs, one gets the identities In the absence of P -odd couplings, these relations fix F V and F A in terms of the vector and axial-vector masses and, moreover, imply that M A > M V . This mass hierarchy remains valid if F 2 V < F 2 V and F 2 A < F 2 A , which is a reasonable working assumption that we will adopt. Notice that Eq. (12) must have the same sign. We will also assume that the inequality M A > M V is fulfilled in all scenarios, even when the 2nd WSR does not apply. The predictions obtained when only P -even operators are considered 5 are shown in Table III (most of these results can be found in Ref. [4]). The more general results, including contributions from both P -even and P -odd op- erators, are displayed in Table IV. In order to ease the notation, and taking into account that most contributions come from resonances in EW triplets and in QCD singlets (R 1 3 ), neither superindices nor subindices are used in this case in Tables III and IV, and from now on; that is, M R ≡ M R 1 3 , unless something else is specified. In Tables III and IV we only show those determinations of LECs that get improved with the short-distance constraints. We consider two types of scenarios: theories where the 2nd WSR applies and a more conservative setting where it does not. The 2nd WSR sharpens considerably the determination of F 1 , and (when only P -even contributions are included) also F 4 and F 6 . In the last two cases, the improvement is lost when the 2nd WSR is dropped, and one gets back to the results in Table I, without further simplification.
In Table IV the determinations are necessarily weaker because more unknown couplings are considered. Nevertheless, we can still obtain useful results assuming that the LECs from odd-parity operators are suppressed with respect to those related to even-parity operators. Assuming F 2 A > F 2 A puts an upper bound on F 1 (indicated with † in Table IV). The assumption | F A G A | < |F V G V | implies F V G V > 0 through Eq. (6), which leads to an upper bound on F 3 (indicated with ‡ ). Finally, assuming | F V λ hV 1 | < |F A λ hA 1 | in Eq. (8), one gets F A λ hA 1 > 0, which implies a lower bound on F 9 (marked with § ).

III. EXPERIMENTAL CONSTRAINTS
Here we summarize the strongest experimental constraints on the EWET LECs from current data: 1. κ W . The ATLAS Collaboration has recently provided the most accurate measurement of the hW W coupling (in SM units), κ W = 1.05 ± 0.04 [19], to be compared with the CMS result, κ W = 1.01 ± 0.07 [20]. These values assume the absence of NP contributions. The naive average of both results gives κ W = 1.04 ± 0.03. A global fit to the LHC Run-1 and Run-2 Higgs data, within the context of the EWET, has determined κ W = 1.01 ± 0.06 [21]. We will adopt this last value, which implies the 95% CL range shown in Table V. 2. c 2V . The ATLAS Collaboration has given the first experimental bound on the hhWW coupling: −1.02 < c 2V < 2.71, at 95% confidence level (CL) [22].
5. F 4 and F 5 . For these LECs we can use the recent analysis of quartic gauge couplings by the CMS collaboration, in the context of the SMEFT [27], which determined |f S0 /Λ 4 | < 2.7 TeV −4 and |f S1 /Λ 4 | < 3.4 TeV −4 (95% CL). Taking into account the relation between these SMEFT LECs and the related EWET LECs [28][29][30], one gets |F 4 | < 0.0006 and |F 5 | < 0.0008 [30], and combining quadratically both bounds, |F 4 + F 5 | < 0.0010. These determinations should be taken with some caution because the experimental analysis has neglected potential uncertainties associated with unitarization effects [30,31]. We will just consider them here to illustrate how stringent become the bounds on the resonance masses, for that level of precision in F 4 and F 5 .
These results are summarized in Table V.

IV. PHENOMENOLOGY
The implementation of short-distance constraints has allowed us to determine some bosonic LECs in terms of very few resonance parameters, as shown in Tables III  and IV. These predictions are plotted in Figures 1 and 2, as functions of the relevant heavy resonance masses. The green bands in Figure 1 indicate the regions allowed by the experimental constraints in Table V. There is still no experimental information available on the LECs plotted in Figure 2.
The top-left panel in Figure 1 displays the dependence of F 1 on M V . For theories where the 1st WSR is obeyed, the dark gray curve indicates the predicted upper bound . Thus, the whole region below this line (gray and brown areas) would be theoretically allowed if only the 1st WSR is assumed. In those scenarios where the 2nd WSR is valid, F 1 is predicted to be a function of M V and M A , the dark gray curve correspond- . This range of M A ∼ M V corresponds actually to the most plausible scenario [4], since the one-loop analysis of the oblique S parameter [6] indicates (under very soft and reasonable assumptions) that κ W ≈ M 2 V /M 2 A when the 2nd WSR is valid, and the LHC experiments have found indeed that κ W ≈ 1.
The WSRs do not play any role in F 3 . If one only considers P -even operators, . This theoretical prediction is shown by the black curve in the top-right panel of Figure 1. Adding possible P -odd contributions, we can only put the upper bound F 3 < −v 2 /(2M 2 A ), which is represented by the same curve but v/c d ). The green area covers the experimentally allowed region, at 95% CL, and it is further extended by a dashed green band that accounts for our estimated one-loop running uncertainties in Eq. (16). If there is a dependence on MV and MA, the gray and/or brown regions cover all possible values for MA > MV . If the 2nd WSR has been considered, it is explicitly indicated in the plot, with the corresponding lines for MA = MV (orange), MA = 1.1 MV (blue), MA = 1.2 MV (red) and MA → ∞ (dark gray). In the case without the 2nd WSR, the theoretically allowed region for F1 is given by the gray and brown regions. In case of using only the even-parity operators, we indicate it in the plot.
this time with M R = M A . Thus, the whole region below this line (gray area) would be allowed in the most general case.
The bottom-left panel in Figure 1 shows the predicted values of Independently of any assumptions concerning WSRs or P -odd operators, the contributions from vector and axial-vector resonance exchanges cancel exactly in the combination Thus, one gets a clean prediction that is shown by the black curve in the bottom-right panel of Figure 1, as function ofM S = M S 1 1 v/c d . A similar cancellation takes place in the combination F 6 + F 7 = d 2 P /(2M 2 P ) ≡ v 2 /(4M 2 P ), which is plotted in the top-right panel of Figure 2, as function ofM P = M P v/d P .
The top-left panel in Figure 2 displays the coupling F 6 , as function of M V , assuming the two WSRs to be valid and considering only P -even operators. The theoretical prediction is always negative and depends on M V , M A and κ W , with its lower bound attained at M A → ∞ (dark gray band). The light-gray area above this bound represents the whole allowed region with M A ≥ M V . The bottom panel in Figure 2. Considering only P -even operators, F 9 = −κ W v 2 /M 2 A , which corresponds to the black band. Its thickness reflects again the experimental error on κ W . When P -odd operators are taken into account, this prediction transforms into a lower bound that is represented by the same band, but this time as function of M R = M V . The whole gray area above the band is then theoretically allowed.
Our tree-level predictions from resonance exchange are actually expected to apply at a high scale around the resonance masses, while the experimental constraints on the LECs in Table V have been obtained at lower energy scales. The one-loop running of the LECs with the renormalization scale is known [32], and the explicit expressions are given in the Appendix, in Eq. (A8). These running contributions are of order 1/(4π 2 ) ∼ 10 −3 and depend on the LO couplings κ W and c 2V . They vanish in the SM limit, κ W = c 2V = 1, as they should. Therefore, for resonances in the few TeV range, M R ∼ 4πv ≈ 3 TeV, we can estimate the potential numerical size of this running effect through the differences Taking into account the current experimental errors on κ W and c 2V , we obtain ∆F 1 = ∆F 3 = 0.9 · 10 −3 , ∆F 4 = 3 · 10 −5 , ∆(F 4 + F 5 ) = 1.7 · 10 −3 , ∆F 6 = 3 · 10 −3 , ∆(F 6 + F 7 ) = 0.6 · 10 −2 , ∆F 9 = 1.4 · 10 −2 . (16) The impact of these running contributions is indicated in Figure 1 with the dashed green bands that enlarge the experimentally allowed regions. Our estimates in Eq. (16) show that the numerical size of the running uncertainty is much more important for the couplings in Figure 2, which are not yet constrained experimentally. This is mainly due to the current poor knowledge on the hhW + W − coupling, since the anomalous dimensions of these LECs are sensitive to c 2V : the uncertainties ∆F 6 , ∆F 6+7 and ∆F 9 in (16) strongly decrease, becoming O(10 −3 ), if we neglect the uncertainty on c 2V . Thus, this error could be sizeably reduced with future data.

V. CONCLUSIONS
Taking into account the great experimental success of the SM, at the currently explored energies, and the emerging evidence about the existence of a mass gap between the SM particles and hypothetical NP states, we have considered a model-independent effective field theory approach to catch any possible deviations from the SM predictions at low energies. Specifically, we have adopted the general non-linear electroweak effective theory (EWET) formalism.
The main aim of this work has been to constrain the scale of the heavy states, which are not directly accessible at current experiments. We have followed a bottom-up approach, where the experimental determination of the LECs of the EWET is used to get imprints of the heavy resonances.
As a consequence, the lightest resonances need to be incorporated in the EWET formalism at higher energies: we have considered here a phenomenological Lagrangian which interpolates between the low-energy and the highenergy regimes [8,9]. In this way, and after integrating out the resonances, the bosonic LECs of the EWET have been determined in terms of resonance parameters, as it is shown in Tables I and II [8,9]. These theoretical predictions can be considerably improved by assuming a proper short-distance behavior of the UV theory, which has allowed us to determine or constrain the bosonic LECs in terms of only resonance masses and the hW W coupling κ W , as it is shown in Tables III and IV. Combining our theoretical predictions with the current experimental bounds on the bosonic LECs, we have obtained the results shown in Figures 1 and 2. These plots push the resonance mass scale to the TeV range, M R ≥ 2 TeV, in good agreement with our previous theoretical estimates in Refs. [4,6], based on a NLO calculation of the S and T oblique parameters within a simplified version of the resonance Lagrangian of Eq. (5).
The oblique S-parameter produces the most precise LEC determination at NLO (F 1 ), which implies the resonance-mass lower bounds M V,A > ∼ 2 TeV, at the 95% CL. On the other hand, the anomalous triple gauge couplings provide a much weaker limit on F 3 , which translates in the softer constraint M V,A > ∼ 0.5 TeV. A recent CMS study has led to very stringent bounds on the couplings that rule WW, WZ and ZZ scattering [27], |F 4,5 | < ∼ 10 −3 in the context of the EWET. In spite of its possible issues regarding unitarity [30,31], it is illustrative to study the implications of such level of precision on the anomalous quartic gauge couplings. The limit on F 4 + F 5 implies that the singlet scalar resonance would have a mass M S 1 1 > ∼ 2 TeV for a S 1 1 W W coupling close to the hW W one (c d ∼ v). Likewise, in the case of BSM extensions with only P-even operators and obeying the two WSRs, the bounds on F 4 constrain the mass of the vector resonance to the range M V > ∼ 2 TeV if M A /M V > 1.1. Nonetheless, lower vector masses would be still allowed if the vector and axial-vector states happened to be very degenerate (1 < M A /M V < 1.1).
Currently, there is no data on the remaining NLO LECs. Triplet pseudo-scalar resonances with masseŝ M P = M P v/d P < ∼ 2 TeV would imply a lower bound F 6 + F 7 > ∼ 5 · 10 −3 , a LEC combination related with W W → hh at NLO. Likewise, a triplet vector resonance with mass M V ∼ 2 TeV leads to the constraint F 6 < −2 · 10 −3 (also relevant for W W → hh scattering) for M A /M V > 1.1, in P-even theories with two WSRs. Finally, the coupling F 9 , related to the hW W vertex at NLO, could be O(10 −2 ) in absolute value for M V,A ∼ 2 TeV (notice the negative sign in Fig. 2). However, one-loop corrections introduce corrections of a similar size in F 6 , F 6 + F 7 and F 9 , respectively, due to the poor knowledge on the hhW W LO coupling c 2V . Any further progress on these three LECs requires a similar improvement in the c 2V precision and the incorporation of one-loop EWET contributions in these experimental analyses.
In summary, the experimental LHC constraints start already to be competitive. This type of analysis will generate much more precise information, once the expected high-statistics data samples from the upgraded LHC runs will be available. ACKNOWLEDGMENTS SU (2) L and SU (2) R matrix fields,Ŵ µ andB µ respectively, and their field-strength tensors are introduced: The covariant derivatives are provided by these fields: The identification [35] explicitly breaks the chiral symmetry group G while preserving the SU (2) L ⊗ U (1) Y gauge symmetry, as in the SM.
Once the fermion doublets are considered, the fieldŝ G µ andX µ are introduced to keep the covariance under local SU (3) C and U (1) X transformations, respectively. The definitions of these fields, their field-strength tensorŝ G µν andX µν and the identifications required to break G while preserving the SU (3) C ⊗ SU (2) L ⊗ U (1) Y gauge symmetry can be found in Ref. [8].
The explicit breaking of custodial symmetry can be incorporated by means of a right-handed spurion: The identification allows one to obtain the custodial symmetry breaking operators induced through quantum loops with internal B µ lines. The power counting of chiral dimensions adopted to organize the operators of the EWET can be summarized as: h ∼ O p 0 , u µ , ∂ µ , T ∼ O p 1 and f ± µν ,Ĝ µν ,X µν ∼ O p 2 [8,9].
The O(p 4 ) operators in Eq. (3) renormalize the UV divergences from one-loop diagrams with LO vertices. The running of the renormalized parameters F i and F i ,