How to use the Standard Model effective field theory

We present a practical three-step procedure of using the Standard Model effective field theory (SM EFT) to connect ultraviolet (UV) models of new physics with weak scale precision observables. With this procedure, one can interpret precision measurements as constraints on a given UV model. We give a detailed explanation for calculating the effective action up to one-loop order in a manifestly gauge covariant fashion. This covariant derivative expansion method dramatically simplifies the process of matching a UV model with the SM EFT, and also makes available a universal formalism that is easy to use for a variety of UV models. A few general aspects of RG running effects and choosing operator bases are discussed. Finally, we provide mapping results between the bosonic sector of the SM EFT and a complete set of precision electroweak and Higgs observables to which present and near future experiments are sensitive. Many results and tools which should prove useful to those wishing to use the SM EFT are detailed in several appendices.


Introduction
The discovery of a Standard Model (SM)-like Higgs boson [1,2] is a milestone in particle physics. Direct study of this boson will shed light on the mysteries surrounding the origin of the Higgs boson and the electroweak (EW) scale. Additionally, it will potentially provide insight into some of the many long standing experimental observations that remain unexeplained (see, e.g., [3]) by the SM. In attempting to answer questions raised by the EW sector and these presently unexplained observations, a variety of new physics models have been proposed, with little clue which-if any-Nature actually picks.
It is exciting that ongoing and possible near future experiments can achieve an estimated per mille sensitivity on precision Higgs and EW observables [4][5][6][7][8][9]. This level of precision provides a window to indirectly explore the theory space of BSM physics and place constraints on specific UV models. For this purpose, an efficient procedure of connecting new physics models with precision Higgs and EW observables is clearly desirable.
In this paper, we make use of the Standard Model effective field theory (SM EFT) as a bridge to connect models of new physics with experimental observables. The SM EFT consists of the renormalizable SM Lagrangian supplemented with higher-dimension interactions: (1.1) In the above, Λ is the cutoff scale of the EFT, O i are a set of dimension d i operators that respect the SU (3) c × SU (2) L × U (1) Y gauge invariance of L SM , and c i are their Wilson coefficients that run as functions c i (µ) of the renormalization group (RG) scale µ. The estimated per-mille sensitivity of future precision Higgs measurements justifies truncating the above expansion at dimension-six operators.
It is worth noting that the SM EFT parameterized by the c i of Eq. (1.1) is totally different from the widely used seven-κ parametrization (e.g., [10]), which captures only a change in size of each of the SM-type Higgs couplings. In fact, the seven κ's parameterize models that do not respect the electroweak gauge symmetry, and hence, violate unitarity. As a result, future precision programs can show spuriously high sensitivity to the κ. The SM EFT of Eq. (1.1), on the other hand, parameterizes new physics in directions that respect the SM gauge invariance and are therefore free from unitarity violations. 1 In an EFT framework, the connection of UV models 2 with low-energy observables is accomplished through a three-step procedure schematically described in Fig. 1. 3 First, the UV model is matched onto the SM EFT at a high-energy scale Λ. This matching is performed order-by-order in a loop expansion. At each loop order, c i (Λ) is determined such that the S-matrix elements in the EFT 1 Equation (1.1) is a linear-realization of EW gauge symmetry. An EFT constructed as a non-linear realization of EW gauge symmetry is, of course, perfectly acceptable. 2 In this work we take "UV model" to generically mean the SM supplemented with new states that couple to the SM.
In particular, the UV model does not need to be UV complete; it may itself be an effective theory of some other, unknown description. 3 For an introduction to the basic techniques of effective field theories see, for example, [11]. The leading order solution to these RG equations is determined by the anomalous dimension matrix γ ij . Finally, we use the effective Lagrangian at µ = m W to compute weak scale observables in terms of the c i (m W ) and SM parameters of L SM . We refer to this third step as mapping the Wilson coefficients onto observables.
In the rest of this paper we consider each of these three steps-matching, running, and mappingin detail for the SM EFT. In the SM EFT, the main challenge presented at each step is complexity: truncating the expansion in (1.1) at dimension-six operators leaves us with O(10 2 ) independent deformations of the Standard Model. 4 This large number of degrees of freedom can obscure the incredible simplicity and utility that the SM EFT has to offer. One of the main purposes of the present work is to provide tools and results to help a user employ the SM EFT and take advantage of the many benefits it can offer.
A typical scenario that we imagine is one where a person has some UV model containing massive BSM states and she wishes to understand how these states affect Higgs and EW obsevables. With a UV model in hand she can, of course, compute these effects using the UV model itself. This option sounds more direct and can, in principle, be more accurate since it does not require an expansion in powers of Λ −1 . However, perfoming a full computation with the UV model is typically quite involved, especially at loop-order and beyond, and needs to be done on a case-by-case basis for each UV model. Among the great advantages of using an EFT is that the computations related to running and mapping, being intrinsic to the EFT, only need to be done once; in other words, once the RG evolution and physical effects of the O i are known (to a given order), the results can be tabulated for general use.
Moreover, for many practical purposes, a full computation in the UV model does not offer considerable improvement in accuracy over the EFT approach when one considers future experimental resolution. The difference between an observable computed using the UV theory versus the (truncated) EFT will scale in powers of E obs /Λ, typically beginning at (E obs /Λ) 2 , where E obs ∼ m W is the energy scale at which the observable is measured. The present lack of evidence for BSM physics coupled to the SM requires in many cases Λ to be at least a factor of a few above the weak scale. With an estimated per mille precision of future Higgs and EW observables, this means that the leading order calculation in the EFT will rapidly converge with the calculation from the UV model, providing essentially the same result for Λ (several × E obs ). 5 For the purpose of determining the physics reach of future experiments on specific UV models-i.e. estimating the largest values of Λ in a given model that experiments can probe-the EFT calculation is sufficiently accurate in almost all cases.
As mentioned above, the steps of RG running the O i and mapping these operators to observables are done within the EFT; once these results are known they can be applied to any set of {c i (Λ)} obtained from matching a given UV model onto the SM EFT. Therefore, an individual wishing to study the impact of some UV model on weak scale observables "only" needs to obtain the c i (Λ) at the matching scale Λ. We put "only" in quotes because this step, while straightforward, can also be computationally complex owing to the large number of operators in the SM EFT.
A large amount of literature pertaining to the SM EFT already exists, some of which dates back a few decades, and is rapidly growing and evolving. Owing to the complexity of the SM EFT, many results are scattered throughout the literature at varying levels of completeness. This body of research can be difficult to wade through for a newcomer (or expert) wishing to use the SM EFT to study the impact of BSM physics on Higgs and EW observables. We believe an explication from a UV perspective, oriented to consider how one uses the SM EFT as a bridge to connect UV models with weak-scale precision observables, is warranted. We have strived to give such a perspective by providing new results and tools with the full picture of matching, running, and mapping in mind. Moreover, our results are aimed to be complete and systematic-especially in regards to the mapping onto observables-as well as usable and self-contained. These goals have obviously contributed to the considerable length of this paper. In the rest of this introduction, we summarize more explicitly our results in order to provide an overview for what is contained where in this paper.
In section 2, we present a method to considerably ease the matching of a UV model onto the SM EFT. The SM EFT is obtained by taking a given UV model and integrating out the massive BSM states. The resultant effective action is given by (1.1), where the higher dimension operators are suppressed by powers of Λ = m, the mass of the heavy BSM states. Although every O i respects SM gauge invariance, traditional methods of evaluating the effective action, such as Feynman diagrams, require working with gauge non-invariant pieces at intermediate steps, so that the process of arranging 5 For example, in considering the impact of scalar tops on the associated Zh production cross-section at an e + e − collider, Craig et. al. recently compared [12] the result of a full NLO calculation versus the SM EFT calculation. They found that the results were virtually indistinguishable for stop masses above 500 GeV. In their calculation, they used Wilson coefficients previously obtained by us in [13]; in section 2 of this paper we explain the details of how these Wilson coefficients are easily computed using the covariant derivative expansion. an answer back into the gauge invariant O i can be quite tedious. Utilizing techniques introduced in [14,15] and termed the covariant derivative expansion (CDE), we present a method of computing the effective action through one-loop order in a manifestly gauge-invariant manner. By working solely with gauge-covariant quantities, an expansion of the effective action is obtained that immediately produces the gauge-invariant operators O i of the EFT and their associated Wilson coefficients.
At one-loop order, the effective action that results when integrating out a heavy field Φ of mass m is generally of the form where D 2 = D µ D µ with D µ a gauge covariant derivative and U (x) depends on the light, SM fields. The typical method for evaluating the functional trace relies on splitting the covariant derivative into its component parts, D µ = ∂ µ − iA µ with A µ a gauge field, and performing a derivative expansion in ∂ 2 − m 2 . This splitting clearly causes intermediate steps of the calculation to be gauge noncovariant. Many years ago, Gaillaird found a transformation [14] that allows the functional trace to be evaluated while keeping gauge covariance manifest at every step of the calculation, which we derive and explain in detail in section 2. In essence, the argument of the logarithm in Eq. (1.2) is transformed such that the covariant derivative only appears in a series of commutators with itself and U (x). The effective action is then evaluated in a series of "free propagators" of the form (q 2 − m 2 ) −1 with q µ a momentum parameter that is integrated over. The coefficients of this expansion are the commutators of D µ with itself and U (x) and correspond to the O i of the EFT. Thus, one immediately obtains the gauge-invariant O i of the effective action. In our discussion, we clarify and streamline certain aspects of the derivation and use of the covariant derivative expansion of [14,15]. Moreover, we generalize the results of [14,15] and provide explicit formulas for scalars, fermions, and massless as well as massive vector bosons. Additionally, a method for obtaining the tree-level effective action in a covariant derivative expansion is presented. While this tree-level evaluation is very straightforward, to the best of our knowledge, it has not appeared elsewhere in the literature.
We believe the CDE to be quite useful in general, but especially so when used to match a UV model onto the SM EFT. The reason for this is that the results obtained using the CDE are, under a few assumptions, universal and can be applied to a large number of UV models, allowing one to calculate one-loop matched Wilson coefficients with ease. To demonstrate this, we compute the Wilson coefficients of a handful of non-trivial examples that could be relevant for Higgs physics, including an electroweak triplet scalar, an electroweak scalar doublet (the two Higgs doublet model), additional massive gauge bosons, and several others.
In section 3 we consider the step of running Wilson coefficients from the matching scale Λ to the electroweak scale m W where measurements are made. Over the past few years, the RG evolution of the SM EFT has been investigated quite intensively [16][17][18][19][20][21][22]. It is a great accomplishment that the entire one-loop anomalous dimension matrix within a complete operator basis has been obtained [17][18][19][20], 6 as well as components of γ ij in other operator bases [21,22]. As the literature has been quite 6 Not only is the computation of γij practically useful, its structure may be hinting at something deep in regards to thorough on the subject, we have little new to contribute in terms of new calculations; instead, our discussion on RG running primarily concerns determining when this step is important to use and how to use it. Since future precision observables have a sensitivity of O(0.1%)-O(1%), they will generically be able to probe new physics at one-loop order. RG evolution introduces a loop factor; therefore, as a rule of thumb, RG running of the c i (Λ) to c i (m W ) is usually only important if the c i (Λ) are tree-level generated. RG evolution includes a logarithm which may serve to counter its loop suppression; however, from v 2 /Λ 2 ∼ 0.1%, we see that Λ can be probed at most to a few TeV, so that the logarithm is not large, log(Λ/m W ) ∼ 3. We note that this estimate also means that in a perturbative expansion, a truncation by loop-order counting is more appropriate than by logarithm power counting.
A common theme in the SM EFT is the choice of operator basis. We will discuss this in detail in section 3, but we would like to comment here on relevance of choosing an operator basis to each step of the matching, running, and mapping steps. One does not need to choose an operator basis at the stage of matching a UV model onto the effective theory. The effective action obtained by integrating out some massive modes will simply produce a set of higher-dimension operators. One can then decide to continue to work with this UV generated operator set as it is, or to switch to a different set due to some other considerations. An operator basis needs to be picked once one RG evolves the Wilson coefficients using the anomalous dimension matrix γ ij , as the anomalous dimension matrix is obviously basis dependent. When the running is relevant, one can choose any operator set compatible with the RG analysis to proceed with it.
In section 4 we consider the mapping step, i.e. obtaining Higgs and EW precision observables as functions of the Wilson coefficients at the weak scale, c i (m W ). While there have been a variety of studies concerning the mapping of operators onto weak-scale observables in the literature [12,19,21,22,[24][25][26][27][28][29][30][31][32][33], to the best of our knowledge, a complete and systematic list does not exist yet. In this paper, we study a complete set of the Higgs and EW precision observables that present and possible near future experiments can have a decent 1% or better sensitivity on. These include the seven Electroweak precision observables (EWPO) S, T, U, W, Y, X, V up to p 4 order in the vacuum polarization functions, the three independent triple gauge couplings (TGC), the deviation in Higgs decay widths {Γ h→ff , Γ h→gg , Γ h→γγ , Γ h→γZ , Γ h→W W * , Γ h→ZZ * }, and the deviation in Higgs production cross sections at both lepton and hadron colliders {σ ggF , σ W W h , σ W h , σ Zh }. We write these precision observables up to linear power and tree-level order in the Wilson coefficients c i (m W ) of a complete set of dimension-six CP-conserving bosonic operators 7 shown in Table 4. Quite a bit calculation steps are also listed in Appendix B. These include a list of two-point and threepoint Feynman rules (appendix B.1) from operators in Table 4, interference corrections to Higgs decay widths (appendix B.2) and production cross sections (appendix B.3), and general analysis on residue modifications (appendix B.4) and Lagrangian parameter modifications (appendix B.5). With a primary interest in new physics that only couple with bosons in the SM, we have taken the Wilson coefficients of all the fermionic operators to be zero while calculating the mapping results. However, renormalization and effective actions [23]. 7 In this paper, we use the term "bosonic operators" to refer to the operators that contain only bosonic fields, i.e. Higgs and gauge bosons. Other operators will be referred to as "fermionic operators". the general analysis we present for calculating the Higgs decay widths and production cross sections completely apply to fermionic operators.

Covariant derivative expansion
In this paper, we advocate the use of the Standard Model EFT from a UV perspective. Let's recapitulate this program. First, match a given UV theory onto the EFT: integrate out heavy physics from the UV model to obtain the Wilson coefficients of the higher dimesion operators in the EFT. Second, run the Wilson coefficients down to weak scale using their RG equations. Third, use the EFT at the weak scale to calculate the contribution of new physics, in the form of non-zero Wilson coefficients, to physical observables. In this section, we present tools that considerably ease the step of matching the UV model onto the EFT. We take up the task of running and mapping in later sections. The process of matching the UV theory onto the EFT is done order-by-order in perturbation theory. As present and future tests of the Standard Model Higgs and gauge sector are typically only sensitive to one-loop order effects, for most purposes it is sufficient to do this matching only up to one-loop order. To this order, the contribution of the UV physics to the low-energy effective action consists of a tree level piece and a one-loop piece.
The point of this section is to present a method for computing the one-loop effective action that leaves gauge invariance manifest at every step of the calculation. By this we mean that one only works with gauge covariant quantities, such as the covariant derivative. We find it somewhat surprising that this method-developed in the 80s by Gaillard [14] (see also her summer school lectures [34] and the work by Cheyette [15])-is not widely known considering the incredible simplifications it provides. Therefore, in order to spread the good word so to speak, we will explain the method of the covariant derivative expansion (CDE) as developed in [14,15]. Along the way, we will make more rigorous and clear a few steps in the derivation, present a more transparent expansion method to evaluate the CDE, and provide generalized results for scalars, fermions, and massless as well as massive gauge bosons. We also show how to evaluate the tree-level effective action in manifestly gauge-covariant manner. In order to explicitly demonstrate the utility of the CDE, we take up a handful of non-trivial examples and compute their Wilson coefficients in the SM EFT.
Besides providing an easier computational framework, the CDE illuminates a certain universality in computing Wilson coefficients from different UV theories. This occurs because individual terms in the expansion split into a trace over internal indices (gauge, flavor, etc.) involving covariant derivatives times low energy fields-these are the operators in the EFT-times a simple momentum integral whose value corresponds to the Wilson coefficient of the operator. The UV physics is contained in the specific form of the covariant derivatives and low energy fields, but the momentum integral is independent of these details and therefore can be considered universal.
So far our discussion has been centered around the idea of integrating out some heavy mode to get an effective action, to which we claim the CDE is a useful tool. More precisely, the CDE is a technique for evaluating functional determinants of a generalized Laplacian operator, det[D 2 + U (x)], where D is some covariant derivative. Therefore the technique is not limited to gauge theories; in fact, the CDE was originally introduced in [14] primarily as a means for computing the one-loop effective action of non-linear sigma models. In these applications, the use of the CDE keeps the geometric structure of the target manifold and its invariance to field redefinitions manifest [14]. Moreover, functional determinants are prolific in the computation of the (1PI or Wilsonian) effective action to one-loop order. Therefore, the use of the CDE extends far beyond integrating out some heavy field and can be used as a tool to, for example, renormalize a (effective) field theory or compute thermal effects.
The 1980s saw considerable effort in developing methods to compute the effective action with arbitrary background fields. While we cannot expect to do justice to this literature, let us provide a brief outline of some relevant works. The CDE developed in [14,15] built upon the derivative expansion technique of [35,36]. A few techniques for covariant calculation of the one-loop effective action were developed somewhat earlier in [37]. While these techniques do afford considerable simplification over traditional methods, they are less systematic and more cumbersome than the CDE presented here [14]. In using a heat kernel to evaluate the effective action, a covariant derivative expansion has also been developed, see, e.g., [38]. This method utilizes a position space representation and is significantly more involved than the approach presented here, where we work in Fourier space.
An outline for this section is as follows. In Sec. 2.1 we consider the tree and one-loop contributions to the effective action in turn and show how to evaluate each using a covariant derivative expansion. The tree-level result is very simple, as well as useful, and, to the best of our knowledge, has not been appeared in the literature before. In Sec. 2.2 we examine evaluation of the functional trace at the more abstract matrix level, thereby clarifying a few steps in the derivation of the CDE. These results are somewhat tangential towards our main focus and can be safely omitted in a first reading. The explicit extension to fermions and gauge bosons is provided in Sec. 2.3 together with summary formulas of the CDE for different spin particles. In Sec. 2.4 we demonstrate how to explicitly evaluate terms in the CDE. Following this, universal formulas for terms in the expansion are presented. As a first example using these results, we derive the β function for non-abelian gauge theory and present the Wilson coefficients for the purely gluonic dimension six operators for massive spin 0, 1/2, and 1 particles transforming under some representation of the gauge group. The universal formulas can also immediately be used to obtain the one-loop effective action for a wide variety of theories, as we show in Sec. 2.5 with a variety of explicit examples. The examples considered are non-trivial demonstrations of the power of the CDE; moreover, they are models that may be relevant to Higgs and other BSM physics: they are related to supersymmetry, extended Higgs sectors, Higgs portal operators, little Higgs theories, extra-dimensional theories, and kinetic mixing of gauge bosons.
We have strived to make accessible the results of this section to a wide audience, primarily because we believe the CDE and its results to be so useful for practical and presently relevant computations. In doing so, however, this section is quite long and it may be helpful to provide a readers guide of sorts in addition to the above outline. Readers mainly interested in the basic idea of the CDE can consider reading the first section, Sec. 2.1, then looking over the universal results in Sec. 2.4 (and equation (2.55) in particular), and skimming a few of the examples in Sec. 2.5.

Setting up the problem
Consider Φ to be a heavy, real scalar field of mass m that we wish to integrate out. Let S[φ, Φ] denote the piece of the action in the full theory consisting of Φ and its interactions with Standard Model fields φ. The effective action resultant from integrating out Φ is given by (2.1) The above defines the effective action at the scale µ ∼ m, where we have matched the UV theory onto the effective theory. In the following we do not write the explicit µ dependence and it is to be implicitly understood that the effective action is being computed at µ ∼ m. Following standard techniques, S eff can be computed to one-loop order by a saddle point approximation to the above integral. To do this, expand Φ around its minimum value, Expanding the action around this minimum, , so that the effective action is given by 3) The first term in the above is the tree-level piece when integrating out a field, i.e. solving for a field's equation of motion and plugging it back into the action, while the second term is the one-loop piece. As is clear in the defining equation of the effective action, Eq. (2.1), the light fields φ are held fixed while the path integral over Φ is computed. The φ(x) fields are therefore referred to as background fields. The fact that the background fields are held fixed while only Φ varies in Eq. (2.1) leads to an obvious diagramatic interpretation of the effective action: the effective action is the set of all Feynman diagrams with φ as external legs and only Φ fields as internal legs. The number of loops in these diagrams correspond to a loop expansion of the effective action.

Covariant evaluation of the tree-level effective action
First, we show how to evaluate the tree-level piece to the effective action in a covariant fashion. The most naïve guess of how to do this turns out to be correct: in the exact same way one would do a derivative expansion, one can do a covariant derivative expansion.
To have a tree-level contribution to the effective action there needs to be a term in the UV Lagrangian that is linear in the heavy field Φ. We take a Lagrangian, where B(x) and U (x) are generically functions of the light fields φ(x) and we have not specified the interaction terms that are cubic or higher in Φ. To get the tree-level effective action, one simply solves the equation of motion for Φ, and plugs it back into the action. The equation of motion for Φ is If the covariant derivative were replaced with the partial derivative, P 2 = −∂ 2 , one would evaluate the above in an inverse-mass expansion producing a series in ∂ 2 /m 2 . The exact same inverse-mass expansion can be used with the covariant derivative as well to obtain 10 In general, the mass-squared matrix need not be proportional to the identity, so that 1/m 2 should be understood as the inverse of the matrix m 2 . In this case, 1/m 2 would not necessarily commute with U and hence we used the matix expansion from Eq. (2.19) in the above equation.
Plugging Φ c back into the Lagrangian gives the tree-level effective action. Using the linearized solution to the equation of motion, Eq. (2.5), we have (2.7) 9 Aµ = A a µ T a with T a in the representation of Φ. We do not specify the coupling constant in the covariant derivative. Of course, the coupling constant can be absorbed into the gauge field; however, unless otherwise stated, for calculations in this paper we implicitly assume the coupling constant to be in the covariant derivative. The primary reason we have not explicitly written the coupling constant is because Φ may carry multiple gauge quantum numbers. For example, if Φ is charged under SU (2)L × U (1)Y then we will take Dµ = ∂µ − igWµ − ig ′ Y Bµ. 10 This is trivially true. In the case of a partial derivative, −∂ 2 − m 2 − U (x), the validity of the expansion relies not only on ∂ 2 /m 2 ≪ 1 but also on U (x)/m 2 ≪ 1, i.e. momenta in the EFT need to be less than m which also means the fields in the EFT need to be slowly varying on distance scales of order m −1 . Obviously, the same conditions can be imposed on the covariant derivative as a whole.
Although we have not specified the interactions in Eq. (2.4) that are cubic or higher in Φ, one needs to also substitute Φ c for these pieces as well, as indicated in the above equation. The first few terms in the inverse mass expansion are (2.8)

CDE of the one-loop effective action
Now let us discuss the one-loop piece of the effective action. Let Φ be field of mass m that we wish to integrate out to obtain a low-energy effective action in terms of light fields. Assume that Φ has quantum numbers under the low-energy gauge groups. The one-loop contribution to the effective action that results from integrating out Φ is where c s = +1/2, +1, or − 1/2 for Φ a real scalar, complex scalar, or fermion, respectively. 11 ,12 We evaluate the trace in the usual fashion by inserting a complete set of momentum and spatial states to arrive at together with the fact that we can bring the e ±iq·x into the logarithm, we see that the P µ → P µ + q µ . Then, after changing variables q → −q, the one-loop effective action is given by Following [14,15], we sandwich the above by e ±P µ ∂/∂q µ In the above it is to be understood that the derivatives ∂/∂q and ∂/∂x ⊂ P act on unity to the right (for e −P ·∂/∂q ) and, by integration by parts, can be made to act on unity to the left (for e P ·∂/∂q ). Since the derivative of one is zero, the above insertion is allowed. We emphasize that the ability to insert 11 The reason fermions have cs = −1/2 instead of the usual −1 is because we have squared the usual argument of the logarithm, ∆Seff = − i 2 Tr log(i / D + . . . ) 2 , to bring it to the form in Eq. (2.9). See Appendix A.1 for details. 12 A vector boson has the same cs as a real scalar. It just has more spin degrees of freedom. A Fadeev-Popov ghost, relevant when integrating out gauge bosons, has the same cs as a fermion by construction. e ±P ·∂/∂q in Eq. (2.13) does not rely on cyclic property of the trace: the "tr" trace in Eq. (2.13) is over internal indices only and we therefore cannot cyclically permute the infinite dimensional matrices in Eq. (2.13). 13 One advantage of this choice of insertion is that it makes the linear term in P µ vanish when transforming the combination (P µ − q µ ), and so the expansion starts from a commutator [P µ , P ν ] = −G ′ µν which is the field strength. Indeed, by making use of the BCH formula and the fact L P ·∂/∂q q µ = [P · ∂/∂q, q µ ] = P µ , we get and similarly Bringing the e ±P ·∂/∂q into the logarithm to compute the transformation of the integrand in Eq. (2.13), one gets the results obtained in [14,15] where we have defined The commutators in the above correspond to manifestly gauge invariant higher dimension operators: In Eq. (2.17a) the commutators of P 's with [D ν , D µ ] = −iG νµ , where G νµ is the gauge field strength, correspond to higher dimension operators of the field strength and its derivatives. In Eq. (2.17b), the commutators will generate higher dimension derivative operators on the fields inside U (x). 13 While the above arguments leading to Eq. (2.13) are correct, they may seem slightly unclear because we have, in fact, brushed over some subtle steps: Why could we use the BCH formula in Eq. (2.12)? Where does this magical unity on the right and left come from? In section 2.2 we provide a more abstract and general treatment that answers these questions and makes clear what transformations in general we can make on the argument of the trace.
While it should be clear, it is worth emphasizing that x and ∂/∂x commute with q and ∂/∂q, i.e. P = i∂/∂x + A(x) and U (x) commute with q and ∂/∂q. This, together with the fact that the commutators in Eq. (2.17) correspond to higher dimension operators, allows us to develop a simple expansion of Eq. (2.16) in terms of higher dimension operators whose coefficients are determined from easy to compute momentum integrals, which we now describe.
Instead of working with the logarithm, we work with its derivative with respect to m 2 . Using ∂ µ to denote the derivative with respect to q, ∂ µ ≡ ∂/∂q µ , and defining ∆ ≡ (q 2 − m 2 ) −1 , the effective Lagrangian is (2.18) In the above, ∆ is a free propagator for a massive particle; we can develop an expansion of powers of ∆ and its derivatives (from the q derivatives inside G and U ) where the coefficients are the higher dimension operators. The derivatives and integrals in q are then simple, albeit tedious, to compute and correspond to the Wilson coeffiecient of the higher dimension operator. Explicitly, using we have (using obvious shorthand notation) There are two points that we would like to draw attention to: Power counting Power counting is very transparent in the expansion in Eq. (2.20). This makes it simple to identify the dimension of the operators in the resultant EFT and to truncate the expansion at the desired order. For example, the lowest dimension operator in G µν is the field strength [D µ , D ν ] = −iG µν ; each successive term in G increases the EFT operator dimension by one through an additional P α . The dimension increase from additional P 's is compensated by additional q derivatives which, by acting on ∆, increase the numbers of propagators.
Universality When the mass squared matrix m 2 is proportional to the identity then ∆ commutes with the matrices in G and U . In this case, for any given term in the expansion in Eq. (2.20), the q integral trivially factorizes out of the trace and can be calculated separately. Because of this, there is a certain universality of the expansion in Eq. (2.20): specifics of a given UV theory are contained in P µ and U (x), but the coefficients of EFT operators are determined by the q integrals and can be calculated without any reference to the UV model.
Before we end this section, let us introduce a more tractable notation that we use in later calculations and results. We provide the notation here for the reader who wishes to skim ahead to results.
As we already have used, ∂ µ ≡ ∂/∂q µ . The action of the covariant derivative on matrix is defined as a commutator and we use as shorthand P µ A ≡ [P µ , A]. We also define G ′ µν ≡ [D µ , D ν ]. 14 To summarize and repeat ourselves: Finally, as everything is explicitly Lorentz invariant, we will typically not bother with raised and lowered indices. With this notation, G and U as defined in Eq. (2.17) are given by

General considerations
Here we look at the covariant evaluation of the one-loop effective action at the operator level, to clarify a few steps presented in the derivation of the previous section. These results are not essential to the rest of this paper and can be omitted in a first reading. For the one-loop effective action, we are interested in evaluating the functional trace, Φ=Φc ≡ Tr f. δ 2 S/δΦ 2 , and hence f , is Hermitian. 15 Since f is Hermitian, its eigenvectors lie in a Hilbert space. Since we are working in a Hilbert space, we will use notation familiar from quantum mechanics. Unfortunately, we cannot diagnolize f and compute its spectrum in general because f depends on arbitrary functions (the background fields). However, we can still develop a perturbative approximation of the trace. For our purposes, f derives from a Lagrangian and is therefore a function of the position and momentum operators, f ( x µ , q µ ).For example, a particular form of f of interest to us in this work is For notational simplicity, in the following we will typically not write the Lorentz indices explicitly.
14 If Dµ = ∂/∂x µ − igAµ, then G ′ µν is related to the usual field strength as G ′ µν = [Dµ, Dν ] = −igGµν. In the case where we have integrated out multiple fields with possibly multiple and different gauge numbers, it is easier to just work with Dµ, hence the definition of G ′ µν . 15 The usual care should be taken when defining the functional determinant: we go to Euclidean space and take K ≡ δ 2 SE/δΦ 2 to be Hermitian, positive definite. For general background fields the matrix is non-singular, although specific field configurations may make K singular, in which case the zero eigenvalues have to be handled with care. These properties allow us to define the functional determinant, det K, as well as the functional trace Tr log K where the Hermiticity of log K follows from that of K. We assume there is no issue with Wick rotation and work in Minkowski space.
To explicitly evaluate Tr f , we will need to give a representation to f . Recall that the operators take on a given representation when acting on some basis vector, e.g. 16 Note that the derivative acts on the eigenvalue of the basis vector which gave the operator that particular representation.
To evaluate Trf we begin by inserting the identity and resolving the identity in momentum space, As before, dq ≡ d 4 q/(2π) 4 , dx ≡ d 4 x, and the lower case "tr" denotes a trace over internal indices only. For the rest of this subsection we will leave the trace on internal indices implicit and drop the "tr" in expressions.
The momentum states |q can be written in a particularly useful way. Define the unit function in x-space as |1 x ≡ dy |y . (2.25) Since a constant function has zero momentum, obviously the unit function in x-space is equivalent to the zero momentum state: |1 x = dp |p p|1 x = dp dy |p e ip·y = dp |p δ(p) = |0 q .
While we could just work with the zero momentum state |0 q , when explicitly evaluating the functional determinant it will be conceptually more convenient to think of it as the unit function |1 x . This state possesses the following properties which are easily checked With the use of the unit function, the plane wave |q can be written as This is easily seen by using the eigen-decomposition e −iq· x = dy e −iq·y |y y|, or even more simply by noting that Eq. (2.27) is obviously consistent with x|q = e −iq·x . Using the decomposition for the momentum states in Eq. (2.27), the trace in Eq. (2.24) is By making use of the Baker-Cambell-Hausdorff (BCH) formula, we see that q → q + q in Eq. (2.28) Inserting a complete set of position states, and taking Taking f as in Eq. (2.2), we see that we recover (2.12) where now it is clear why we could use the BCH formula to get (2.12). Moreover, it is explicitly clear what it means for the derivative i∂ x to be acting on unity to the right; in the above q takes a representation from x|, x| q = i∂ x x|, and acts upon the eigenvalue of x|. When x| hits |1 x , x|1 x = 1, it is to be understood that the derivative i∂ x then acts on unity when it gets all the way to the right. Let us consider more general transformations that can be made within the inner product of Eq. (2.28). Note that since q is simply a parameter, it commutes with everything. Let us promote this parameter to a second momentum operator, q → q 2 , that acts on a second position-momentum space. Denoting the original x and q as x 1 and q 1 , the commutation relations are (2.31) x 2 and q 2 are operators on a second Hilbert space; the entire Hilbert space is the direct product H = H 1 ⊗ H 2 . We denote states in H 1 ⊗ H 2 with a single bra or ket with a semi-colon separating labels between the H i and the state in H 1 always to the left of the semi-colon. For example, Making use of the property where |1 q i = dp i |p i is the unit function in q i -space, we see that that we can rewrite the trace in Eq. (2.28) as where in the last line we used BCH to shift q 1 → q 1 + q 2 . What have we gained by going through this more abstract way of writing the trace? The point is that Eq. (2.34) makes it clear that we can make many transformations on f ( x 1 , q 1 + q 2 ) that leave the trace invariant: A large number of operators leave the unit function |1 x 1 ; 1 q 2 invariant; by inserting these into the inner product in Eq. (2.34) we can then regard them as transformations on f ( x 1 , q 1 + q 2 ). Moreover, by promoting the parameter q to be operator valued, q → q 2 , it is clear that we can consider transformations on q 2 as well. The idea, of course, is that some of these transformations may bring f to a particularly convenient form.
Let us consider the operators which leave the state |1 x 1 ; 1 q 2 invariant. Let h( x i , q i ) be an analytic function of the position and momentum operators and we ask We have put h in the exponential for convenience, from which clearly the above condition is satisfied when h annihilates |1 x 1 ; 1 q 2 . We are not particularly interested in general considerations on the form of h, but rather concern ourselves with pointing out some classes of h that satisfy Eq. (2.35) which will prove useful in explicit calculations. Recalling that q |1 x = x |1 q = 0, we see that if h only depends on q 1 and x 2 then any function h( q 1 , x 2 ) such that h( q 1 , 0) = 0 or h(0, x 2 ) = 0 will annihilate |1 x 1 ; 1 q 2 . If we consider h to depend on x 1 as well, then any function h( x 1 , q 1 , x 2 ) such that h( x 1 , q 1 , 0) = 0 will annihilate |1 x 1 ; 1 q 2 . This follows from that fact that since x 2 commutes with x 1 and q 1 , we can always bring it to the right where it will annihilate |1 q 2 .
Let h( x i , q i ) and h ′ ( x i , q i ) be two Hermitian operators satisfying Eq. (2.35). We can therefore insert these into the inner product in Eq. (2.34) and consider the properties of the transformed operator (2.36) When h ′ = h, this amounts to a unitary transformation on f . In this case, assuming f has a welldefined Taylor expansion, we have e ih f ( x 1 , q 1 + q 2 )e −ih = f e ih x 1 e −ih , e ih ( q 1 + q 2 )e −ih (2.37) and the transformations can be evaluated using the BCH formula Eq. (2.29). When h is not very complicated, these are not hard to compute. As an example, consider the case h = − q 1 · x 2 : This transformation takes us from the starting point of the derivative expansion as described around Eq. (2.30) to the form used in [35,36]. Finally, let us consider the case where f contains the covariant derivative: From the above discussion we have, We consider the unitary transformation e ih with h = − P 1 · x 2 , which is the operator statement of the transformation introduced by [14] and used in the previous subsection in deriving the CDE. As per our discussion on the allowed forms of h, while P 1 does not annihilate |1 x 1 , x 2 does annihilate |1 q 2 and therefore h |1 x 1 ; 1 q 2 = 0. The nice property of this h is that it shifts q 2 by the covariant derivative: q 2 → q 2 − P 1 + . . . where the higher order terms are commutators of the covariant derivative with itself times powers of x 2 , i.e., just as in Eq. (??). Upon using this shift and inserting the complete set of states, into Eq.(2.39), it is straightforward to see that we recover the covariant derivative expansion in formula (2.16).

CDE for fermions, gauge bosons, and summary formulas
The CDE as presented in section 2.1.2 is for evaluating functional determinants of the form where P µ = iD µ is a covariant derivative. As such, the results of section 2.1.2 apply for any generalized Laplacian operator of the form −P 2 + W (x). 17 The lightning summary is where we G and W are given in Eq. (2.22) with U replaced by W and we are using the notation defined in Eq. (2.21). In section 2.1.2 we took W (x) = m 2 + U (x) for its obvious connection to massive scalar fields. When we integrate out fermions and gauge bosons, at one-loop they also give functional determinants of generalized Laplacian operators of the form −P 2 + W (x). It is straightforward to apply the steps of section 2.1.2 to these cases. Nevertheless, it is useful to tabulate these results for easy reference; therefore, in this section we summarize the results for massive scalars, massive fermions, and massless gauge fields. We relegate detailed derivations of the fermion and gauge boson results to appendix A.1. The results for fermions were first obtained in [14] 18 and for gauge bosons in [15]. 17 This is loosely speaking, but applies to many of the cases physicists encounter. More correctly, the functional determinant should exist and so we actually work in Euclidean space and consider elliptic operators of the form +P 2 + W (x) with W hermitian, positive-definite. The transformations leading to the CDE in section 2.1.2 then apply to these elliptic operators as well. In the cases we commonly encounter in physics, these properties are satisfied by the fact that operator is the second variation of the Euclidean action which is typically taken to Hermitian, positive-definite. 18 We note that there is an error in the results for fermions in [14].
Let us state the general result and then specify how it specializes to the various cases under consderation. The one-loop effective action is given by where the constant c s and the form of U depend on the species we integrate out, as we explain below. After evaluating the trace and using the transformations introduced in [14] and explained in section 2.1.2, the one-loop effective Lagrangian is given by where the lower case trace, "tr", is over internal indices and

Real scalars
The effective action originates from the Gaussian integral For this case, in Eqns. (2.41) and (2.42) we have

Complex scalars
The effective action originates from the Gaussian integral For this case, in Eqns. (2.41) and (2.42) we have

Massive fermions
We work with Dirac fermions. The effective action originates from the Gaussian integral where / P = γ µ P µ with g µ the gamma matrices. As shown in appendix A.1, in Eqns. (2.41) and (2.42) we have where σ µν = i[γ µ , γ ν ]/2 and, by definition, Note that the trace in (2.42) includes tracing over the spinor indices. Besides the σ · G ′ term in U ferm , all other terms in U ferm and the −P 2 term are proportional to the identity matrix in the spinor indices which, since we use the 4 × 4 gamma matrices, is the 4 × 4 identity matrix 1 4 .

Massless gauge fields
We take pure Yang-Mills theory for non-abelian gauge group G, where t a G are generators in the adjoint representation and µ(G) is the Dynkin index for the adjoint representation. 19 We are considering the 1PI effective action, Γ[A], of the gauge field A µ .
We explain the essential details here and explicate them in full in appendix A.1. The 1PI effective action is evaluated using the background field method: the gauge field is expanded around a background piece and a fluctuating piece, A µ (x) = A B,µ (x) + Q µ , and we integrate out Q µ . The field Q µ is gauge-fixed in such a way as to preserve the background field gauge invariance. The gauge-fixed functional integral we evaluate is, where c a are Fadeev-Popov ghosts. In the above, (2.47) The second piece is the from the gauge field Q a µ which gives Eqns. (2.41) and (2.42) with With m 2 = 0, Eqns. (2.41) and (2.42) contain IR divergences. These IR divergences can be regulated by adding a mass term for Q a µ and c a (essentially keeping m 2 in Eqns. (2.41) and (2.42)). 19 For representation R, the Dynkin index is given by tr 20 Note the similarity with the fermion case, where σ µν /2 is the generator of Lorentz transformations on spinors. Explicitly, the components of J µν are given by (J µν )ρσ = i(δ µ ρ δ ν σ − δ µ σ δ ν ρ ).

Massive vector bosons
We consider a UV model with gauge group G that is spontaneously broken into H which is the SM gauge group. A set of gauge bosons Q i µ , i = 1, 2, ..., dim(G) − dim(H) that correspond to the broken generators obtain mass m Q through Higgs mechanism by "eating" the Nambu-Goldstone bosons χ i . These heavy gauge bosons form a representation of the unbroken gauge group (SM gauge group). The general Lagrangian up to quadratic term in Q i µ , which we derive in appendix A.1, is where D µ denotes the covariant derivative of the SM gauge symmetry, M µν parameterize a generic interaction term, and we have adopted a general R ξ gauge with c i andc i denoting the ghosts. One can compute the effective action by evaluating

Evaluating the CDE and universal results
In the present subsection we explicitly show how to evaluate terms in covariant derivative expansion of the one-loop effective action in Eqns. (2.18) and (2.20). Following this, we provide the results of the expansion through a given order in covariant derivatives. Specifically, for an effective action of the form S eff ∝ Tr log(−P 2 + m 2 + U ) we provide the results of the CDE through dimension-six operators assuming U is at least linear in background fields. These results make no explicit reference to a specific UV model, and therefore they are, in a sense, universal. This universal result is tabulated in Eq. (2.55) and can be immediately used to compute the effective action of a given UV model.

Evaluating terms in CDE
Let us consider how to evaluate expansion terms from the effective Lagrangian of Eq. (2.18), which we reproduce here for convenience In the above, G and U are as defined in Eq.
, and we employ the shorthand notation defined in (2.21). We also used the fact that we define the integrals The effective action from a given I n integral is given by ∆L In = −ic s I n . G νµ and U are an infinite expansions in covariant derivatives of G ′ νµ and U , and thus contain higher-dimension operators (HDOs). Therefore, each I n is an infinite expansion containing these HDOs. For this work, motivated by present and future Higgs observables, we are interested in corrections up to dimension-six operators. This dictates how many I n we have to calculate as well as what order in G νµ and U we need to expand within a given I n .
As a typical example to demonstrate how to evaluate the I n , we consider I 1 , This term is fairly easy to compute and captures the basic steps to evaluate any of the I n while also highlighting a few features that are unique to low order terms in the expansion. We remind the reader that q µ and ∂ µ commute with P µ and U , which is what makes the I n very simple to compute. We also assume that the mass-squared matrix m 2 commutes with G ′ µν and U . 21 In this case, ∆ commutes with the HDOs in G and U , i.e. [∆, P α 1 . . . P αn G ′ µν ] = 0 and similarly for the HDOs in U . This allows us to seperate the q-integral from the trace over the HDOs.
Let us now evaluate I 1 in (2.54). We consider the U term first, 21 This is always the case if m 2 is proportional to identity, i.e. if every particle integrated out has the same mass. If we integrate out multiple particles with different masses, typically m 2 commutes with G ′ µν but, in general, will not commute with U . For m 2 to commute with G ′ µν , in the operator P 2 −m 2 −U (x), it amounts to assuming Pµ and m 2 are block diagnol of the form Pµ = diag(P (1) µ , . . . , P (n) µ ) and m 2 = diag(m 2 1 , . . . , m 2 n ). Physically, this means we are integrating out n particles, where the ith particle has mass-squared m 2 i and a covariant derivative P (i) µ associated to its gauge interactions. The block-diagnol mass matrix means we diagnolized the mass matrix before integrating out the particles. If U happens to have the same block-diagnol structure, then of course m 2 commutes with U as well.
Recall that the covariant derivative action on a matrix is defined as the commutator, e.g. P α U = [P α , U ]. Since the trace of a commutator vanishes, everything but the n = 0 term in the above vanishes. Equivalently, the n ≥ 1 terms are total derivatives and therefore do not contribute to the effective action. Thus, The above term is divergent. It may be the case-as in the above integral-that the order of integration does not commute and changes the divergent structure of the integral. In these cases, to properly capture the divergent structure (and therefore define counter-terms) the integral on m 2 should be performed first since we are truly evaluating dq dm 2 ∂ ∂m 2 tr log(. . . ). 22 In this paper, we use dimensional regularization with MS for our renormalization scheme, in which case where µ is the renormalization scale.
We now turn our attention to the pieces in I 1 involving G µν . The term linear in G in I 1 vanishes since it is the trace of a commutator, as was the case for the higher derivative derms in U discussed above. Thus, only the G 2 term in non-zero and we seek to evaluate We evaluate the above up to dimension-six operators.
where we dropped the O(P 5 ) terms since they vanish as required by Lorentz invariance. It is straightforward to plug the above back into I 1 and compute the q-derivatives and integrals. For example, the G ′2 ∂ 2 requires computing 22 Simple power counting easily shows that divergences in In can only occur for n = 0, 1, and 2. In the expansions of G and U within I0,1,2, it is not difficult to see that there are only four non-vanishing divergent terms: I0, in I1 they are the tr U and tr G ′ µν G ′ ρσ terms, and in I2 it is the tr U 2 term.
where we computed the m 2 integral first and used dimensional regularization with MS. Thus, we see that which we clearly recognize as a contribution to the β function of the gauge coupling constant. The other O(P 6 ) terms in the expansion of G 2 are computed similarly. In appendix A.2 we tabulate several useful identities that frequently occur, such as ∂ n α 1 ...αn ∆ and what this becomes under the q-integral. For example, in the above computation we used The end result of computing the q-integrals for the O(P 6 ) terms in I 1 gives There are only two possible dimension-six operators involving just P µ and G ′ µν , namely tr Using the Bianchi identity and integration by parts, tr [A(P µ B)] = −tr [(P µ A)B]+ total deriv., the above can be arranged into just these two dimension-six operators: Combining all these terms together, we find the contribution to the effective Lagrangian from I 1 is For the reader following closely, we note that the only contribution to tr (G ′ µν ) 2 is the above term from I 1 , while tr (P µ G ′ µν ) 2 and tr G ′3 also receive contributions from I 2 . In a similar fashion, one can compute the other I n . In the next section we tabulate the result of all possible contributions to dimension-six operators from the I n ; in appendix A.3 the results for each individual I n are listed.

Universal results
We just showed how to evaluate terms in the CDE to a given order. Here we tabulate the results that allow one to compute the one-loop effective action through dimension six operators. In the next subsection we use these results to obtain the dimension-six Wilson coefficients of the SM EFT for several non-trivial BSM models.
The one-loop effective action is given by where, as discussed these in section 2.3, c s and U (x) depend on the species we integrate out. We assume that the mass-squared matrix m 2 commutes with U and G ′ µν . Under this assumption, we tabulate results of the CDE through dimension-six operators. In general, U may have terms which are linear in the background fields. 23 In this case, although the scaling dimension of U is two, its operator dimension may be one. Simple power counting tells us that we will have to evaluate terms in the I n integrals of Eq. (2.4.1) through I 6 . 24 In appendix A.3, we give the result of this calculation for each of the relevant terms in I 1 -I 6 . Gathering all of the terms together, the one-loop effective action is: 23 For example, a Yukawa interaction yφψψ for massive fermions leads to a term linear in the light field φ: from Eq. (2.46), Uferm ⊃ 2mM (x) = ymφ. 24 While this is tedious, it isn't too hard. Moreover, there are many terms within each In that we don't need to compute since they lead to too large of an operator dimension. For example, the only term in I6 that we need to compute is All other terms in I6 have to large of operator dimension and can be dropped.
-25 - • This formula is the expansion of a functional trace of the form ic s Tr log − P 2 + m 2 + U (x) where P µ = iD µ is a covariant derivative and U (x) is an arbitrary function of spacetime. We have worked in MInkowski space and defined the one-loop action and Lagrangian from ic s Tr log − P 2 + m 2 + U = ∆S eff,1-loop = d 4 x∆L 1-loop .
• The results of Eq. (2.55) are valid when the mass-squared matrix m 2 commutes with U (x) and • The lower case "tr" in (2.55) is over internal indices. These indices may include gauge indices, Lorentz indices (spinor, vector, etc.), flavor indices, etc..
• c s is a constant which relates the functional trace to the effective action, á la the first bullet point above. For example, for real scalars, complex scalars, and grassman valued fields c s = 1/2, 1, and −1, respectively. U (x) is a function of the background fields. In section 2.3 we discussed the form of U (x) for various particle species, namely scalars, fermions, and gauge bosons.
• Given the above statements, it is clear that (2.55) is universal in the sense that it applies to any effective action of the form Tr log − P 2 + m 2 + U . 25 For any specific theory, one only needs to determine the form of the covariant derivative P µ and the matrix U (x) and then (2.55) may be used. We provide several examples in the next subsection.
• Equation (2.55) is an expansion of the effective Lagrangian through dimension-six operators. U has scaling dimension two, but its operator dimension may be one or greater. In the case U contains a term with unit operator dimension, one needs all the terms in (2.55) to capture all dimension-six operators.
• The lines proportional to m 4 , m 2 , and m 0 in (2.55) come from UV divergences in the evaluation of the trace; µ is a renormalization scale and we used dimensional regularization and MS • The lines proportional to m 2 and m 0 can always be absorbed by renormalization. They can also be used to find the contribution of the particles we integrate out to the β-functions of operators.

Evaluation of the pure glue pieces
The operators involving only gauge bosons, G ′2 at dimension four and (P G ′ ) 2 and G ′3 at dimension six, are determined solely by stating the field content and their representations under the gauge groups. As such, we can evaluate these terms more generally. For the dimension four term G ′2 we will immediately produce the β function of Yang-Mills coupling constant.
For simplicity, we take a simple gauge group 26 and evaluate the contribution of different particle species to these pure glue operators. In this case, the covariant derivative is given by where G µν is the Yang-Mills field strength. All particle species contribute to renormalization of the Yang-Mills kinetic term, −(G a µν ) 2 /4, through the tr G ′2 µν term in (2.55). In addition, the magnetic moment coupling for fermions and gauge bosons is contained within U , U ⊃ −iM µν G ′ µν where M µν is the Lorentz generator in a given representation-see Eqns. (2.46) and (2.48). This term then contributes to the Yang-Mills kinetic term through tr U 2 . Evaluating these terms for a particle with spin j particle and representation R under the gauge group we have where d(j) is the number of components of the spin j particle 27 and µ(R) is the Dynkin index of the R th representation. 28 And for the tr U 2 term we have 25 Under the assumption m 2 commutes with U and G ′ µν ; see the second bullet point. 26 For a semi-simple group, the following results apply to each individual gauge group. 27 d = 1, 4, and 4 for scalars, Dirac fermions, and vectors, respectively. 28 where k = 1 (k = 2) for Dirac spinors (vectors). 29 Combining these terms together, we see that a given species that we integrate out produces We recognize the term in square brackets as the contribution to the one-loop β function coefficient. 30 In particular, for scalars, fermions, and vector bosons (including the ghost contribution, Eq. (2.47)), we have Dirac fermions In a similar fashion, we can compute the dimension-six pure glue operators. In Eq. (2.55), these come from tr(P µ G ′ µν ) 2 and tr (G ′ µν G ′ νσ G ′ σµ ) as well as tr U 3 and tr(P µ U ) 2 when U contains the magnetic moment coupling. These traces are straightforward to compute. Defining the dimension-six operators Adding these terms up we have In Table 1 we tabulate these coefficients for different species, where in the massive gauge boson case, proper contributions from Goldstone and ghosts are already included. The particle has mass m and transforms in the R th representation of the group, with µ(R) its index. Real scalars are half the value of complex scalars.

Example calculations
Here we give several example models where we calculate the effective action using the covariant derivative expansion. Our main purpose is to demonstrate how to use the CDE for both tree-level and one-loop level calculations; however, we also have picked models that are of phenomenological interest. Since we are using CDE to do matching step between UV models and the SM EFT, it should be understood that all the Wilson coefficients obtained in this section are at the matching scale Λ, namely that all our results are actually about c i (Λ). And we drop the specification of RG scale throughout the section. The examples include different types of species and come at varying levels of complexity. This serves two purposes. In the arena of pedagogy, we hope to show how details are handled for actual computations a reader might be interested in performing. In the arena of demonstrating the utility of the CDE, the more complicated the example, the more the CDE saves over traditional methods.
Before going into the examples, we would also like to make a list of possible UV models that have tree-level contributions to the effective action. As mentioned before, in order to do so, the UV Lagrangian must contain a term that is linear in the heavy field being integrated out. If we require the UV model to be renormalizable and the heavy field being integrated out only couples with the bosonic fields of the SM, then there are only five of them: The above list exhausts the possibilities of tree-level Wilson coefficients from renormalizable UV models in which heavy states only couple with the bosonic fields of the SM. First, any heavy fermionic field Ψ would need a fermionic field from the SM to form a Lagrangian term that is linear in it, so it immediately goes out of the scope of the consideration here. Second, a heavy vector boson has to be gauged in order for the UV theory to be renormalizable. And by requirement of gauge invariance and dimension counting, one can easily see that the No.5 above is the only possibility. Now the only case left is a heavy scalar Φ that couples to the SM Higgs H and gauge bosons. But a renormalizable interaction term between Φ and the SM gauge bosons cannot be linear in Φ due to gauge invariance. Therefore what is left for us is to count all the possible Lagrangian terms formed by Φ and H that is linear in Φ. After appropriate diagonalization of Φ and H, we do not need to consider the quadratic terms. Then there are only two types of renormalizable interactions H a H b Φ ab and H a H b H c Φ abc , where we have written the SM Higgs field H in terms of its four real components H a with a = 1, 2, 3, 4. Because only symmetric combinations are non-vanishing, it is clear that there are in total 10 real components Φ ab that are enumerated by No.1 and No.2 in the above list, and 20 real components Φ abc that are enumerated by No.3 and No.4.

Electroweak triplet scalar
Let us consider an electroweak triplet scalar Φ with neutral hypercharge. The Lagrangian contains the trilinear interaction H † ΦH, where H is the electroweak Higgs doublet. This interaction, being linear in Φ, leads to a tree-level contribution to the effective action when we integrate out Φ.
While our main purpose here is to demonstrate how to use the CDE, we note that triplet scalars are phenomenologically interesting and well studied. As shown below, the electroweak T parameter is generated at tree-level due to the custodial violating interaction H † ΦH. The strong constraints on the T parameter require the triplet scalar to have a large mass, m ≫ v. In this regime, the leading terms of the EFT are quite accurate.
For readers interested in comparing the CDE with traditional Feynman diagram techniques, we note that triplet scalars were studied within the EFT framework in [39] where the Wilson coefficients 31 The H 3 below is only a sketchy form, whose exact expressions are as follows were calculated using Feynman diagrams (see the appendices of [39]). Tree-level Feynman diagrams involving scalar propagators are pretty straightforward to deal with; yet, we believe that even in this simple case the CDE offers a significantly easier method of calculation. In particular, at no point do we (1) have to break the Lagrangian into gauge non-covariant pieces to obtain Feynman rules, (2) look up a table of higher dimension operators to know how to rearrange the answer back into a gauge-invariant form, or (3) consider various momenta configurations of external particles in order to extract which particular higher dimension operator is generated.

Tree-level matching
Let Φ = Φ a T a be an electroweak, real scalar triplet with hypercharge Y Φ = 0. 32 We take the SU (2) L generators in the fundamental representation, T a = τ a = σ a /2 with σ a the Pauli matrices. The Lagrangian involving Φ and its interactions with the Standard Model Higgs doublet is given by 33 The interaction H † ΦH, being linear in Φ, leads to a tree-level contribution to the effective action. To calculate this contribution, we follow the steps outlined in section 2.1.1. Introducing an obvious vector notation and writing the Lagrangian as in Eq. (2.4), Now we need to evaluate the terms in the above. For the B · B term we have 35 from which it follows B a U B a = 2ηκ 2 |H| 6 . 32 For YΦ = 0, Φ a must be complex. Only for YΦ = 0 or 1 can Φ have a trilinear interaction with H. 33 The coupling names and normalization are chosen to coincide with those in [39]. 34 At tree-level, the quartic ( Φ· Φ) 2 term contributes only to dimension eight and larger operators, and we therefore ignore it presently. The quartic piece does lead to a contribution of the one-loop effective action, which is discussed below. 35 Here and below we use the following relation for generators T a in the fundamental representation of SU (N ): Integrating by parts, the term in involving the covariant derivative is Squaring this, using the identity in the previous footnote and the one in Eq. (A.52) we have Table 4. Putting it all together, we find where O 6 = |H| 6 . As mentioned previously, these results were also obtained in [39] using Feynman diagrams. 36 The first term in the above can be absorbed into the renormalization of the Higgs quartic coupling. As discussed in section 4, O T contributes to the electroweak T parameter. Thus, we see in the effective theory that the T parameter is generated at tree-level.

One-loop level matching
Let us also calculate the one-loop effective action from integrating out the scalar triplet. It is given by 36 The notation in the first reference of [39] uses the three operators O1, O2, and O ′ T where we added the prime since it is not the same as our OT . What they call O ′ T is now more commonly called OHD. where 1 3 is the 3 × 3 identity matrix and we explicitly wrote it above to remind the reader that each piece in U ′ is a matrix. The term in square brackets above is due to the fact that there is a non-zero treelevel piece, i.e. that Φ c = 0. Diagrammatically, this term leads to one-loop graphs that are connected but not 1PI in Φ. As we will see below, at leading order this term will lead to a renormalization of |H| 4 and a logarithmically divergent contribution to |H| 6 (see Fig. 2 for diagrams). To leading order, we can take the lowest order solution of Φ c in Eq. (2.68), namely With this, U ′ is given by where U = 2η |H| 2 as in Eq. (2.67).
To evaluate the one-loop effective action, we take the universal results from Eq. (2.55) with c s = 1/2 since Φ a is a real scalar. As U ′ contains no term that is linear in fields, for dimension six and less operators we take the m 2 , m 0 , and m −2 terms from Eq. (2.55) where the generators t a G are in the adjoint representation. 38 Keeping only up to dimension-six operators and using the operator definitions given in table 4, the traces 37 Note that these terms can be used to find the contribution of Φ to the beta functions of SM couplings. In particular, the triplet contributes to the running of the Higgs' mass and quartic coupling and also to the SU (2)L gauge coupling g. This is easy to see since Plugging these back into (2.70), the dimension-six operators in the one-loop effective action are (2.71)

Extra EW scalar doublet
Here we integrate out an additional electroweak scalar doublet Φ with hypercharge Y Φ = −1/2 and mass m 2 ≫ v 2 . This is essentially the two Higgs doublet model (2HDM) where the mass term for the extra scalar is taken large compared to the symmetry breaking scale. 2HDM models are interesting to consider on their own since they affect the properties of the EWSB sector and therefore might help explain some of the mysterious properties of the Higgs boson. 2HDM also naturally arise in many interesting extensions of the SM, such as supersymmetry.
The general Lagrangian for a 2HDM model can be rather complex; often, if the UV model doesn't already impose some restriction on the 2HDM model (as it does in, e.g., supersymmetry), then some other simplifying approximation is made to make more tractable the study the second doublet. Below, we will consider the most general scalar sector for the second EW doublet; this is rather easy to handle within our EFT framework and requires very little additional effort. 40 The most general Lagrangian consisting of an extra EW scalar doublet Φ with Y Φ = −1/2 39 For example, 40 Of course, a large reason why this is much easier in the EFT framework is because we have made the simplifying assumption that the second doublet is heavy.
interacting with the Higgs sector is given by are the SU (2) L generators in the fundamental representation, and Φ · H = ǫ αβ Φ α H β . The first line of the above is the potential of Φ alone, the second line contains a linear term in Φ which leads to a tree-level contribution to the effective action, while the last line contains interactions with the Higgs doublet H that appear in the effective action at one-loop order.
The main purpose of this section is to show how to use the covariant derivative expansion; in this regard, we remain agnostic as to what types of restrictions specific 2HDM models might impose on the Lagrangian in (2.72). However, let us make a few, brief comments. We have not included a Yukawa sector with couplings to Φ; these would lead to tree-level generated dimension-six operators involving only fermions. If a parity Φ → −Φ, H → H is imposed, then the terms in the second line of (2.72) and extra Yukawa terms are forbidden. This parity prevents Φ from developing a vacuum expectation value 41 and Φ in this case is sometimes known as an "inert Higgs" [40]. Finally, imposing an exact or approximate global U (1) on Φ eliminates the second line in (2.72), the term proportional to λ 3 in (2.72), and any potential Yukawa terms involving Φ.
Before proceeding, it is convenient to define H ≡ iσ 2 H * . This definition makes it easier to see what matrix we are tracing over when we integrate out Φ, Then Φ · H = H † Φ and the Lagrangian in (2.72) is given by Note that obviously | H| 2 = |H| 2 ; we wrote the above purely in terms of H, but we will often write | H| 2 as |H| 2 .

Tree-level matching
When we integrate out the massive doublet the term linear in Φ in (2.72), η H |H| 2 H † Φ + h.c. , leads to a tree-level contribution to the effective action. As this interaction is cubic in the Higgs field, it is simple to see that the only dimension-six operator will be O 6 = |H| 6 . Concretely, B from the general tree-level formula Eq.
and the tree-level effective action through dimension-six operators is (2.75)

One-loop-level matching
Let us now find the one-loop effective action from integrating out the massive scalar doublet Φ in Eq. (2.73). As previously stated, one of the main reasons we are providing these examples is to show how to use the covariant derivative expansion. All the couplings in Eq. (2.73) make the effective action calculation complicated, but not very difficult. For the moment, however, let us make several simplifying assumptions on the couplings simply so that the basic setup and use of the CDE is not obscured. After we show the CDE for the simpler Lagrangian, we will return to the full Lagrangian in Eq. (2.73) and use the CDE to compute the one-loop effective action.

Simpler 2HDM theory
For the simplifying assumptions, let us impose a global U (1) on Φ so that η H = η Φ = λ 3 = 0 in the Lagrangian. Again, we will come back and let these terms be non-zero shorty. In this case, there is no tree-level effective action. We integrate Φ out of the Lagrangian After performing the gaussian integral we are left with the effective action where we have explicitly denoted the 2 × 2 identity matrix by 1 2 . Therefore, In trG ′3 the anti-symmetry on the Lorentz indices only leaves trW 3 non-vanishing. Thus, 42 where µ(R) is the Dynkin index for representation R and is equal to 1/2 for the fundamental representation and O 3W is as defined in table 4.
using trAτ a = λ 2 H † τ a H = −λ 2 H † τ a H and a few other manipulations, it is straightforward to see that

Full 2HDM theory
Now we return to the full Lagrangian in (2.73) and leave all couplings non-zero. This makes the calculation more complicated; however, as is hopefully clear from the above, the calculation will not be too difficult: We will simply need to end up evaluating some traces which, while tedious, is very straightforward. In many regards, most of the work goes into setting up the matrix that we are tracing over. In the present case, with arbitrary couplings in (2.73) and the non-zero tree-level piece Φ c , the matrix is complicated looking, but with a little effort we can see how it all works out.
To evaluate the one-loop effective action, we expand the action around the solution to the equation of motion, Φ = Φ c + σ. For our desired accuracy of dimension-six operators, it is sufficient to take the lowest order solution of Φ c , which, from (2.74), is given by Because the interaction (Φ · H) 2 is holomorphic in Φ, it is easiest to treat Φ and Φ * as separate variables. This is equivalent to splitting Φ into its real and imaginary pieces, although more convenient to work with. Then, upon expanding Φ = Φ c + σ and doing a little algebra, the terms quadratic in σ are 77) 42 We used A few comments: • We are treating σ and σ * as separate variables, which is the same procedure as working with the real and imaginary parts of σ.
• The one-loop effective action is given by with the matrix in (2.77) inserted into the trace. Note the factor of 1/2; we take c s = 1/2 since we are treating σ and σ * as separate, real variables.
• The subscript "c" on ξ c and k c is to remind us that these terms originate from the background configuration Φ c . Note that ξ c ∼ # |H| 2 + # |H| 4 contains powers of the Higgs field. ξ c will only contribute at low orders in the covariant derivative expansion since we are truncating at dimension-six operators.
• Up to dimension-six operators, the background configuration Φ c will only contribute to |H| 4 and |H| 6 . The contribution to the Higgs quartic is absorbed by renormalization. The |H| 6 modifies the Higgs self-coupling, which is currently unobservable. Therefore, if one is not interested in the |H| 6 Wilson coefficient, Φ c can be set to zero for the one-loop computation, saving some calculational effort.
We now use the CDE to compute ∆S eff,1-loop . In the CDE formulas, we take where 1 2 is the 2 × 2 identity matrix. It is clear that the effective action of the form Tr log − P 2 + m 2 + U , so that the transformation e ±P µ ∂/∂q µ in Eq. (2.13) is still allowed and the CDE proceeds as discussed.
Thus, we can immediately use the universal results in Eq. (2.55) with matrices P µ and U defined as above in (2.79), and all that is left to do is evaluate some traces. This is straightforward, albeit slightly tedious. Note that, since ξ c ∼ O(|H| 2 ), we only need to keep ξ c for tr U and tr U 2 . When ξ c = 0, the matrix A ′ in (2.78) and (2.79) is equal to the matrix A defined in Eq. (2.76). Tabulating only dim-6 operators, using the operator definitions in tbl 4, and including a factor of 1/2 for convenience, we find (2.80) Plugging these traces into Eq. (2.55) we obtain the one-loop effective Lagrangian. We summarize the results below.

Electroweak scalar doublet summary
We took an electroweak scalar doublet Φ with hypercharge Y Φ = −1/2 and Lagrangian and integrated out Φ to find the dimension-six operators of the effective action matched at one-loop order.
The tree-level effective action, given in Eq. (2.75), only contains O 6 = |H| 6 . The one-loop effective action is obtained from plugging the traces in Eq. (2.80) into (2.55). This piece contains a host of dimension-six operators that affect electroweak and Higgs physics. In summary, the effective Lagrangian at the matching scale is given by where the Wilson coefficients are given in Table 2.

Supersymmetry and light scalar tops
Supersymmetric states at or near the electroweak scale could explain the origin of this scale and its radiative stability. Scalar tops (stops) hold a priveledged position in providing a natural explanation to origin of the EW scale. This motivated us in a previous work [13] to study the low-energy EFT that results when stops are integrated out. In that work, we considered a supersymmetric spectrum with light stops and other superpartners decoupled and computed the Wilson coefficients of the one-loop effective action. In this paper, we provide more details of how the Wilson coefficients are calculated using the covariant derivative expansion.
As stops carry all SM gauge quantum numbers, every operator in Table 4 is generated. In [13], we computed the Wilson coefficients separately using the CDE and traditional Feynman diagram techniques. The results agreed, providing a good consistency check of the calculation. 43 More importantly, however, the two methods highlighted just how much effort the CDE saves over traditional techniques. No doubt the CDE computation is still complicated, as we will see below, but that is because stops have a large number of various interactions with the SM Higgs and gauge bosons. Nevertheless, it is extremely systematic.
We integrate out the multiplet φ = (Q 3 ,t R ) T , the Lagrangian of which up to quadratic order is given by where (2.83) 43 A recent paper by Nathaniel Craig et al. [12] computed the correction from scalar tops to the associate production cross section σ Zh . They calculated both using the UV Lagrangian up to 1-loop order perturbation, and using our Wilson coefficients in EFT. Excellent agreement was found, which also serves as a good consistency check. and the matrix U is Now with both the representation and the interaction matrix U at hand, we are ready to make use of Eq. (2.55) to compute the Wilson coefficients. However, in order for Eq. (2.55) to be valid, we need U to commute with the mass square matrix m 2 , which limits us to the generate mass scenario It is also worth noting that due to the appearance of X t , U is no long quadratic in H, but also contains linear term of H. This means that one has to keep up to O(U 6 ) in computing the Wilson coefficients of dimension-six operators. Another thing to keep in mind while evaluating the terms in Eq. (2.55) is thatQ 3 andt R have different charges under the SM gauge group, and the covariant derivative D µ or P µ = iD µ should take on the appropriate form for each.
Through a straightforward, albeit tedious use of Eq. (2.55), we obtain the final result of Wilson coefficients listed in Table 3.

A real singlet scalar
In this example, we consider a heavy real singlet scalar field Φ with mass m that couples to the SM through the following Lagrangian Table 3. Wilson coefficients c i for the operators O i in Table 4 generated from integrating out MSSM stops with degenerate soft mass mt. g s , g, and g ′ denote the gauge couplings of SU (3), SU (2) L , and U (1) Y , respectively, h t = m t /v, and tan β = H u / H d in the MSSM.
Integrating out this heavy scalar Φ generates a SM EFT that matches the above UV theory at the scale Λ = m. Let us first calculate the tree-level contribution to the effective Lagrangian. Following the procedure described in Section 2.1.1, we first obtain the equation of motion which gives the solution Plugging this solution back to Eq. (2.84), we get the tree-level effective Lagrangian This is clearly in a form of Eq. (2.9), with U = k |H| 2 and G ′ µν = [D µ , D ν ] = [∂ µ , ∂ ν ] = 0. Plugging these specific values of U and G ′ µν into Eq. (2.55), we obtain (2.87)

A SU (2) L quartet scalar
In this example, we consider a heavy complex SU (2) L quartet scalar Φ with mass m and SM hypercharge Y Φ = 3 2 . This Φ can couple to the SM through the following Lagrangian where H 1 and H 2 are components of the SM Higgs field H = (H 1 , H 2 ) T . Again, we following the procedure described in Section 2.1.1 to compute the tree-level effective Lagrangian. We first get the equation of motion which gives the solution Useful traces related are Putting them all together into Eq. (2.55), we obtain (2.98)

Kinetic mixing of gauge bosons
In this example, we consider a heavy U (1) gauge boson K µ with mass m K that has a kinetic mixing with the SM U (1) Y gauge boson B µ through where K µν denotes the field strength K µν = ∂ µ K ν −∂ ν K µ . Again, following the procedure described in Section 2.1.1, we first find the equation of motion of this heavy gauge boson K µ which as usual for vector bosons, can be decomposed into two equations solving which we get the classical solution Next we plug this solution back to the UV model Lagrangian (Eq. (2.99)) to get the tree-level effective Lagrangian. With B µν K µν = 2(∂ ν B µν )K µ , we obtain It should be clear from Eq. (2.100) and Eq. (2.101) that when we integrate out a gauge boson at tree-level, its kinetic parts always generate effective operators with higher dimension d > 6, and can hence be ignored from the very beginning. This would bring dramatic simplification to tree-level calculation. For example, the above example can be repeated with this trick as follows. We first rewrite the kinetic mixing interaction term as B µν K µν = 2(∂ ν B µν )K µ , and then drop the kinetic terms of K µ to simplify the Lagrangian Then the equation of motion is much simpler which gives the solution Plug this back to the simplified UV model Lagrangian Eq. (2.102), we get the same result as before This example does not have non-trivial one-loop piece of effective action.

A heavy SU (2) gauge boson
As another example involving heavy gauge bosons, we consider an SU (2) 1 ×SU (2) 2 gauge symmetry where with a scalar Φ transforming as a bifundamental. We take the Standard Model fermions and Higgs field to be localized to the SU (2) 1 gauge group. (We suppress color and hypercharge; the full gauge symmetry is SU The scalar Φ takes a vev, breaking the SU (2) groups down to their diagonal subgroup, which we identify we identify with the weak interactions of the SM, SU (2) 1 × SU (2) 2 → SU (2) L . This is simply a deconstructed [41] version of an extra-dimensional model (e.g. [42]), where the weak gauge bosons, being a diagonal combination of the SU (2) 1 × SU (2) 2 gauge bosons, "propagate in the bulk", while the SM fermions and Higgs only transform under one gauge group and are therefore "localized". The relevant kinetic terms of the Lagrangian are where the scalar Φ transforms as a bifundamental, Φ → U 1 ΦU † 2 . The covariant derivative of the UV theory is given by 44 where g i and A i µ = A a i µ τ a i are the gauge coupling and gauge bosons of the SU (2) i with the generators τ a i taken in the fundamental representation. A vacuum expectation value for Φ, The mass eigenstates are Q a ≡ 1 where W a are the SM gauge bosons corresponding to the unbroken symmetry SU (2) L , and Q a obtain a mass m 2 Q = (g 2 1 + g 2 2 )v 2 /4 from the Higgs mechanism. Q µ tranforms in the adjoint of the unbroken SU (2) L .
In terms of the mass eigenstates, the covariant derivative becomes where τ a L = τ a 1 + τ a 2 are the unbroken generators and we identify g ≡ g 1 g 2 / g 2 1 + g 2 2 as the weak coupling constant of the SM. We expand Φ around Φ , (2.107) 44 Note that the action of Dµ on Φ is DµΦ = ∂µΦ − ig1A1 µΦ + ig2ΦA2 µ where the χ a are the Nambu-Goldstone bosons transforming in the adjoint of the unbroken SU (2) L and h is the massive Higgs field. Expanding out L K in terms of W µ , Q µ , χ, and adding with the gauge fixing piece L gf , the ghost term L gh , as well as the interaction L I between Q µ and the SM we get the full Lagrangian up to quadratic terms in Q a µ as where H denotes the SM Higgs field, and J aµ W is the source current of the SM W a µ . Although the heavy fields Q i µ couple directly to the fermions in SM, as we will see shortly, upon using equation of motion, the tree-level effective Lagrangian turns out to contain only bosonic operators.
At leading order, the solution to equation of motion is (2.115)

Running of Wilson coefficients and choosing operator set
In the last section, we gave a detailed description of the covariant derivative expansion method, a technique to efficiently achieve the first step in Fig. 1 -matching a UV model with a SM EFT. In this section, we move on to the second step -connecting c i (Λ) with c i (m W ). This step is nothing but evolving the Wilson coefficients c i (µ) from the matching scale µ = Λ down to the weak scale µ = m W , according to their renormalization group (RG) equations. At the leading order, these RG equations are governed by the anomalous dimension matrix γ ij : The solution at leading order is which gives So this RG running step seems really straightforward once the anomalous dimension matrix γ ij is known. However, the calculation of γ ij is far from easy. It is quite tedious and depends on the choice of the operator set {O i }. At least, one needs to choose a RG closed set of operators to make Eq. (3.1) valid [17]. Also, there are generically indirect contributions to γ ij in addition to the direct contribution. Here direct contribution refers to the case when O j generates O i directly through a loop Feynman diagram, while indirect contribution means that O j generates some O k outside the operator set chosen, whose elimination (through equation of motion or identity) in turn gives O i . For the rest of this section, we will discuss the topic of choosing operator set in light of doing the RG running analysis. But before going down to that, we would like to point out that although the running of Wilson coefficients is conceptually an important step, there are stringent requirements on the UV models for it to be practically relevant. This is because the near future measurements can at best achieve a precision of per mille level. From v 2 Λ 2 ∼ 0.1%, we also see that Λ can be probed at most up to a few TeV. So the logarithm is not large log Λ m W ∼ 3. Therefore in the perturbative calculation, a truncation by loop order counting is more appropriate than by logarithm power counting. Then the per mille level precision means that we can truncate our perturbative calculations at the one-loop level. It is clear from Eq. (3.3) that the RG running effect is one loop smaller than c j (Λ). So first, if c j (Λ) itself is already of one-loop size from the UV model, then its running effects are of two-loop size and hence practically negligible. Second, even in the case that c j (Λ) is generated at tree level, its one-loop size RG running contribution to c i (m W ) would be subdominant if c i (Λ) is also generated at tree level together with c j (Λ). In summary, practically one needs to take into account of γ ij only when both of the following conditions are satisfied: 1. c j (Λ) is generated at tree level from the UV model.

c i (Λ) is not generated at tree level from the UV model.
As was discussed in Section 2.5, focusing on renormalizable UV models in which new physics only couples with the bosonic fields of the SM, there are only five of them satisfying the condition (1) above. All of them are listed out in Section 2.5. Among these five, the No.3 and No.4 only generate O 6 at tree-level. Since O 6 does not run into other dimension-six operators, clearly the condition (2) above cannot be satisfied by No.3 or No.4. So RG analysis is needed for only the rest three on the list. Of course, if one allows the new physics to couple with SM fermions directly, then more UV models can be found in which the RG analysis is relevant.

Popular operator bases in the literature
A few popular choices of dim-6 operator set are commonly used in the literature (see [29] for a recent review). They have been developed with two different types of motivations: (1) completeness, and (2) phenomenological relevance. In spite of that, however, they are actually not very different from each other. In this subsection, we will briefly describe each basis and then discuss the relation among them.
With a motivation of completeness, one starts with enumerating all the possible dim-6 operators that respect the Standard Model gauge symmetry. A systematic classification of the dim-6 operators would be very helpful in order not to miss any of them. Then many combinations of them are found to be zero due to simple identities. One can remove these redundances and shrink the operator set. In addition, many other combinations are zero upon using equation of motions, and hence would not contribute to physical observables which are on-shell quantities. These combinations can also be removed because they are redundant in respect of describing physics. After all of these reductions, one arrives at an operator set that is non redundant but still complete, in a sense that it has the full capability of describing the physical effects of any dim-6 operators. Clearly, the non redundant complete set of operators form an "operator basis". There are, of course, multiple choices of operator bases, all related by usual basis transformations.
The first attempt of this completeness motivated construction dates back to [43], where 80 dim-6 operators were claimed to be independent. However, it was later discovered that there were still some redundant combinations within the set of 80. The non-redundant basis was eventually found to contain only 59 dim-6 operators [44]. To respect this first success, we will call the 59 dim-6 operators listed in [44] the "standard basis". During the past year, the full anomalous dimension matrix γ ij has been calculated in the standard basis [17][18][19][20]23].
The second type of motivation in choosing operator set is the relevance to phenomenology. With this kind of motivation, one usually starts with a quite small set of operators that are immediately relevant to the physics concerned. But if the RG running effects turns out to be important, a complete operator set is required for the analysis. In that case, one can extend the initial operator set into a complete operator basis by adding enough non redundant operators to it. Popular operator bases constructed along this line include the "EGGM basis" [22], the "HISZ basis" [45], and the "SILH basis" [21,27,46]. These three bases are all motivated by studying physics relevant to the Higgs boson and the electroweak bosons. As a result, they all maximize the use of bosonic  5) and the other five are defined in Table 4. There are two identities among them as following The three phenomenological relevance motivated bases are not that different from the standard basis either. As mentioned before, due to motivation difference, the second type maximizes the use of bosonic operators. It turns out that to obtain the "EGGM basis" from the standard basis, one only needs to do the following basis transformation (trading five fermionic operators into five bosonic operators using equation of motion): (3.12)

Choosing operator set in light of RG running analysis
There are obviously three types of operator sets in general: (1) an exact complete set which is just an operator basis that related to the standard basis by a basis transformation, (2) an over complete set that has some redundant operators, and (3) an incomplete set that lacks of some components compared to a complete operator basis. Let us comment on these three choices one by one.

• Working with a Complete Operator Basis
This case is fairly straightforward. Since the full anomalous dimension matrix γ ij in the standard basis has been computed [17][18][19], one just needs to carry out the basis transformation to obtain the γ ij in the new basis.
• Working with an Over Complete Operator Set Sometimes it is helpful to use a redundant operator set because it can make the physics more transparent. An obvious drawback of working with an over complete set of operators is that the size of γ ij would be larger than necessary, and that the value of γ ij would not be unique. However, this does not necessarily mean that γ ij is harder to calculate. For example, consider the extreme case of using all the dim-6 operators, before removing any redundant combination. This is a super over complete set, and as a result the size of γ ij would be way larger than that in standard basis. But with this choice of operator set, all the contributions to γ ij are direct contributions by definition. Some of these direct contributions would become indirect in a smaller operator set, and one has to accommodate them by using equation of motion or identity, which is a further step of calculation.
• Working with an Incomplete Operator Set An operator basis contains 59 operators, which has 76 (2499) real Wilson coefficients for the number of generation being one (three) [19]. So that is practically a very large basis to work with. In some cases, only a small number of operators are relevant to the physics considered, then it is very tempting to just focus on this small incomplete set, for purpose of simplification. However, while a complete or over operator basis is obviously guaranteed to be RG closed, an incomplete operator set is typically not. When the incomplete operator set is not RG closed, Eq. (3.1) no longer holds. To fix this problem, one can view the incomplete operator set {O i } as a subset of a certain complete operator basis {O i , O a }. Once this full operator basis is specified, one has a clear definition of the sub matrix γ ij to compute the RG induced effects. Obviously, the off diagonal block γ ia is generically nonzero, which means some operator O a outside the chosen operator set {O i } can also be RG induced, which could bring additional constraints on the UV model. The ignorance of this makes the constraints over conservative. (see also the discussion in section 2 of [22]). Table 4. Dimension-six bosonic operators for our mapping analysis.

Mapping Wilson coefficients onto observables
So far we have described how to compute the Wilson coefficients c i (Λ) from a given UV model, and how to run them down to the weak scale c i (m W ) with the appropriate anomalous dimension matrix γ ij . This section then is devoted to the last step in Fig. 1 -mapping c i 45 onto the weak scale precision observables. The Wilson coefficients c i will bring various corrections to the precision observables at the weak scale. The goal of this section is to study the deviation of each weak scale precision observable as a function of c i . It is worth noting that our SM EFT parameterized by Eq. (1.1) and c i is totally different from the widely used seven-κ parametrization (for example see [10]), which parameterizes only a size change in each of the SM type Higgs couplings. The seven-κ actually parameterize models that do not respect the electroweak gauge symmetry and hence violates unitarity. As a result, future precision programs show spuriously high sensitivity on them. Our SM EFT on the other hand, parameterize new physics in the direction that respects the SM gauge invariance and free from unitarity violation.
In order to provide a concrete mapping result, we need to specify a set of operators to work with. With in mind a special interest in UV models in which new physics is CP preserving and couples with the SM only through the Higgs and gauge bosons, we choose the set of dim-6 operators that are purely bosonic and CP conserving. All the dim-6 operators satisfying these conditions are listed in Table 4. This set of effective operators coincides with the set chosen in [22], supplemented by the operators O D and O R . Wilson coefficients of all the fermionic operators are assumed to be zero.
There are four categories of precision observables on which present and near future precision programs will be able to reach a per mille level sensitivity: (1) Electroweak Precision Observables (EWPO), (2) Triple Gauge Couplings (TGC), (3) Higgs decay widths, and (4) Higgs production cross sections. In the mapping calculation, we can keep only up to linear order of Wilson coefficients and we only include tree-level diagrams of the Wilson coefficients. This is because the near future precision experiments would only be sensitive to one-loop physics, and we practically consider each power of 1 Λ 2 c i as one-loop size, since it is already known that the SM is a very good theoretical description and the deviations should be small. Although in some UV models, Wilson coefficients can arise at tree-level, the corresponding 1 Λ 2 must be small enough to be consistent with the current constraints. So considering 1 Λ 2 c i as one-loop size is practically appropriate.

Electroweak precision observables
Electroweak precision observables represent the oblique corrections to the propagators of electroweak gauge bosons. Specifically, there are four transverse vacuum polarization functions: Π W W (p 2 ), Π ZZ (p 2 ), Π γγ (p 2 ), and Π γZ (p 2 ) 46 , each of which can be expanded in p 2 Π(p 2 ) = a 0 + a 2 p 2 + a 4 p 4 + O(p 6 ). Two out of these expansion coefficients are fixed to zero by the masslessness of the photon Π γγ (0) = Π γZ (0) = 0. Another three combinations are fixed (absorbed) by the definition of the three free parameters g, g ′ , and v in electroweak theory. So up to p 2 order, there are three left-over parameters that can be used to test the predictions of the model. These are the Peskin-Takeuchi parameters S, T , and U defined as in [47,49], which captures all the possible non-decoupling electroweak oblique corrections. As higher energy scales were probed at LEP II, it was proposed to include also the coefficients of p 4 terms, which brings us four additional parameters W, Y, X, V [48,50,51]. 46 Throughout this paper, we use Π(p 2 ) to denote the additional part of the transverse vacuum polarization function due to the Wilson coefficients. In a more precise notation, one should use Π new (p 2 ) as in [47] or δΠ(p 2 ) as in [48] for it, but we simply use Π(p 2 ) to reduce the clutter. That being said, our Π(p 2 ) at leading order is linear in ci. So in total, we have seven EWPO parameters in consideration, S, T, U, W, Y, X, V . In this paper, we take the definitions of them as listed in Table 5, 47 where for the purpose of conciseness, we use the alternative set {Π 33 , Π BB , Π 3B } instead of {Π ZZ , Π γγ , Π γZ }. 48 And due to the relation W 3 = c Z Z + s Z A and B = −s Z Z + c Z A 49 , the two set are simply related by the transformation Table 6 summarizes the mapping results of the seven EWPO parameters, i.e. each of them as a linear function (to leading order) of the Wilson coefficients c i . These results are straightforward to calculate. First, we calculate Π W W (p 2 ), Π ZZ (p 2 ), Π γγ (p 2 ), and Π γZ (p 2 ) in terms of c i . This can be done by expanding out the dim-6 operators in Table 4, identifying the relevant Lagrangian terms, and reading off the two-point Feynman rules. The details of these steps together with the results of Π W W (p 2 ), Π ZZ (p 2 ), Π γγ (p 2 ), and Π γZ (p 2 ) (Table 12) are shown in Appendix B.1. Next, we compute the alternative combinations Π W W (p 2 ) − Π 33 (p 2 ), Π 33 (p 2 ), Π BB (p 2 ), Π 3B (p 2 ) using the transformation relation Eq. (4.2)-Eq. (4.4), the results of which are also summarized in Appendix B.1 (Table 13). Finally, we combine Table 13 with the definitions of EWPO parameters (Table 5) to obtain the results in Table 6.
We would like to emphasize the importance of W and Y parameters. It should be clear from the definitions Table 5 that the seven EWPO parameters fall into four different classes: {S, X}, {T, U, V }, {W }, and {Y }. Therefore W and Y out of the four p 4 order EWPO parameters supplement the classes formed by S, T, U (see also the discussions in [51]). Our mapping results ( Table 6) also show that W 47 Our definitions in Table 5 agree with [49] and [51]. Many other popular definitions are in common use as well (e.g. see [7,47,48]). The main differences lie in the choice of using derivatives of Π(p 2 ) evaluated at p 2 = 0, such as Π ′ W W (0) etc., versus using some form of finite distance subtraction, such as etc. Up to p 4 order in Π(p 2 ), this discrepancy would only cause a disagreement in the result of U . For example, the definition in [47] would result in nonzero U parameter from the custodial preserving operator O2W : U = s 4 Z α 4m 2 Z Λ 2 c 2W = 0. In this paper, we stick to the definition in [49] to make U a purely custodial violating parameter. Under our definition, U = 0 at dim-6 level. 48 One may also have a concern that these definitions through the transverse polarization functions Π(p 2 ) are not generically gauge invariant. In principle, these Π(p 2 ) functions can be promoted to gauge invariant ones Π(p 2 ) by a "pinch technique" prescription.
(For examples, see discussions in [28,52,53].) 49 Throughout this paper, we adopt the notation c Z ≡ cos θZ etc., with θZ denoting the weak mixing angle. We do not use θW in order to avoid clash with the Wilson coefficient for the operator OW . and Y are practically more important compared to X and V , for W and Y are nonzero while X and V vanish at dim-6 level.

Triple gauge couplings
The TGC parameters can be described by the a phenomenological Lagrangian [54][55][56] Among the five parameters above, there are two relations due to an accidental custodial symmetry. We take g Z 1 , κ γ , and λ γ as the three independent parameters. The other two can be expressed as [56] The SM values of TGC parameters are g Z 1,SM = κ γ,SM = 1, λ γ,SM = 0. Their deviations from SM are currently constrained at percent level [57], and will be improved to 10 −4 level at ILC500 (see No.2 in [4]). Their mapping results are summarized in Table 7. 50

Deviations in Higgs decay widths
The dim-6 operators bring deviations in the Higgs decay widths from the Standard Model. In this paper, we study all the SM Higgs decay modes that near future linear colliders can have sub-percent sensitivity on, i.e. Γ ∈ Γ h→ff , Γ h→gg , Γ h→γγ , Γ h→γZ , Γ h→W W * , Γ h→ZZ * . Our analysis for the decay modes through off-shell vector gauge bosons h → W W * and h → ZZ * apply to all their fermionic modes, namely that h → W W * → W lν/W dū and h → ZZ * → Zff.
For each decay width Γ above, we define its deviation from the SM ǫ ≡ Γ Γ SM − 1.
(4.8) 50 These results are also obtained in [22]. ǫ hff,I = 0  It turns out that at leading order (linear power) in c i , this deviation is generically a sum of three parts, (1) the "interference correction" ǫ I , (2) the "residue correction" ǫ R , and (3) the "parametric correction" ǫ P : In the following, we will first give a brief description of the meaning and the mapping results of each part, and then explain in detail how to derive these results.

Brief description of the results
• "Interference Correction" ǫ I ǫ I captures the effects of new amputated Feynman diagrams iM AD,new (c i ) introduced by the dim-6 effective operators. This modifies the value of the total amputated diagram Table 9. Residue corrections ǫ R and parametric corrections ǫ P to Higgs decay widths. The results of residue modifications ∆r h , ∆r W , ∆r Z and parameter modifications ∆w g 2 , ∆w v 2 , ∆w s 2 Z , ∆w y 2 f are listed in the appendix.
Upon modulus square, the cross term, namely the interference between the new amplitude and the SM amplitude, gives the leading order contribution to the deviation: where dΠ f denotes the phase space integral, and the overscore denotes any step needed for getting the unpolarized result, namely that a sum of final spins and an average over the initial spins, if any. The results of ǫ I are summarized in Table 8 Table 8  • "Residue Correction" ǫ R ǫ R captures the effects of mass pole residue modifications by the dim-6 effective operators. We know from LSZ reduction formula that the invariant amplitude iM should equal to the value of amputated diagram iM AD multiplied by the square root of the mass pole residue r k of each (4.12) So besides the corrections to iM AD discussed before, a mass pole residue modification ∆r k of an external leg particle k would also feed into the decay width deviation. Upon modulus square, this part of deviation is The results of ǫ R for each decay width are summarized in the second column of Table 9. The values of the relevant residue modifications ∆r k are listed in Appendix B.4 (Table 14). Note that unlike the interference correction ǫ I , the residue correction ǫ R corresponds to a contribution with the size of Γ SM × c i . But for Γ hgg , Γ hγγ , and Γ hγZ , the SM value Γ SM is already of oneloop size. So ǫ hgg,R , ǫ hγγ,R , and ǫ hγZ,R should be one-loop size in Wilson coefficients, namely that 1 16π 2 × c i . In our order of approximation, this size should be neglected for consistency. (4.14) The ǫ I and ǫ R discussed previously only take into account of the explicit dependence on c i , with {ρ} held fixed. The implicit dependence on c i through modifying the Lagrangian parameter ρ is what we call "parametric correction": where ∆w ρ denotes the Lagrangian parameter modification The parametric correction ǫ P in terms of ∆w ρ are summarized in the third column of Table 9. And a detailed calculation of ∆w ρ is in Appendix B.5, with the results summarized in Table 15.
As with the residue correction case, ǫ hgg,P , ǫ hγγ,P , and ǫ hγZ,P are one-loop size in Wilson coefficients and hence neglected for consistency.

Detailed derivation
Clearly from Eq. (1.1), our SM EFT goes back to the SM when all c i = 0. Thus up to linear power of c i , the deviation defined in Eq. (4.8) is Clearly, in addition to the explicit dependence on c i , Γ also has an implicit dependence on c i through the Lagrangian parameters ρ(obs, c i ): Putting it another way, the first term in the above shows the deviation when ρ are fixed numbers. But ρ are not fixed numbers. They are a set of Lagrangian parameters determined by a set of experimental measurements obs through relations that get modified by c i as well. So the truly fixed numbers are the experimental inputs obs. By adding the second piece in Eq. (4.21), we get the full amount of deviation with obs as fixed input numbers. By putting obs in the place of ln Γ, one can also explicitly check that Eq. (4.21) keeps obs fixed. Making use of the fact we clearly see that Because of Eq. (4.21), the deviation Eq. (4.17) is split into two parts where the implicit dependence part is defined as the parametric correction ǫ P (4.25) with the parameter modifications ∆w ρ defined as The explicit dependence part can be further split by noting that Therefore we have with ǫ I and ǫ R defined as So in summary, the total deviation in decay width has three parts ǫ = ǫ I + ǫ R + ǫ P , with where For each decay width in consideration, we computed these three parts of deviation. The results are summarized in Table 8 and Table 9. It is worth noting that this splitting is a convenient intermediate treatment of the calculation, but each of ǫ I , ǫ R , ǫ P alone would not be physical, because it depends on the renormalization scheme as well as the choice of operator basis. It is the total sum of the three that reflects the physical deviation in the decay widths.

Deviations in Higgs production cross sections
The dim-6 operators also induce deviations in the Higgs production cross sections. In this paper, we focus on the production modes σ ∈ σ ggF , σ W W h , σ W h , σ Zh , which are the most important ones for both hadron colliders such as the LHC and future lepton colliders such as the ILC. As with the decay width case, we define the cross section deviation There are again three types of corrections The mapping results are summarized in Table 10 and Table 11. Relevant new amputated Feynman diagrams for ǫ I are listed in Appendix B.3. The calculation of the interference correction to σ W W h turns out to be very involved. Its lengthy analytical expression ǫ W W h,I (s) does not help much, so we instead show its numerical results in Table 10.  Fig. 3. Mathematica code to obtain Fig. 3 is also attached to our paper.

Conclusions
In this paper, we advocate a practical three-step procedure of using the SM EFT to connect UV models with weak scale precision observables. This procedure helps translating precision measurements into constraints on the UV models in concern. For the first step, we gave a detailed explanation of our matching technique between UV models and the SM EFT -the covariant derivative expansion method. With this technique, one can compute the effective action up to one-loop order in a manifestly gauge covariant fashion, and also obtain a universal formalism that is easy to use for different UV models. We showed rigorous derivation of its formalism, discussed its application to integrating out heavy scalars, fermions, and gauge bosons, presented universal formalism for all types of UV models, and calculated the effective Lagrangian for a few specific UV models as explicit examples.
Then about the second step, we showed how to run the Wilson coefficients down from Λ to the weak scale m W , where a closely related topic, choosing the operator set, was also discussed. Finally for the The numerical results of the auxiliary functions in ǫ W W h,I (s) are shown in Fig. 3. Table 11. Residue corrections ǫ R and parametric corrections ǫ P to Higgs production cross sections. The results of residue modifications and parameter modifications are listed in the appendix.

A More on CDE
This appendix shows some details in using the CDE method. First, in appendix A.1, we present some details of the derivation of CDE for fermions and gauge bosons. Appendix A.2 then list out quite a bit useful identities that one frequently encounters while using CDE. Finally, appendix A.3 shows intermediate steps in deriving the universal formula of the CDE.

A.1 Details of CDE for fermions and gauge bosons Fermions
We now consider the functional determinant for massive fermion fields and provide the formulas for the covariant derivative expansion for them. We work in the notation of Dirac fermions, denoting the gamma matrices by γ µ and employing slashed notation, e.g. / D = γ µ D µ . This discussion is easily modified if one wants to consider Weyl fermions and use two-component notation.
Consider the Lagrangian containing the fermions to be where m is the fermion mass and M (x) is in general dependent on the light fields φ(x). Upon integrating over the Grassman valued fields in the path integral, the one-loop contribution to the effective action is given by where, as before, P µ ≡ iD µ . Using Tr log AB = Tr log A + Tr log B and the fact that the trace is invariant under changing signs of gamma matrices we have where G ′ µν ≡ [D µ , D ν ], as defined in Eq. (2.21). We thus see that the trace for fermions, is of the form Tr log(−P 2 + m 2 + U ). Therefore, all the steps in evaluating the trace and shifting by the covariant derivative using e ±P ·∂/∂q are the same as previously considered and we can immediately write down the answer from Eq. (2.16). Defining the one-loop effective Lagrangian for fermions is then given by where G and U ferm are defined as in Eq. (2.22) with U → U ferm .

Massless gauge bosons
Here we consider the one-loop contribution to the 1PI effective action from massless gauge fields. The spirit here is slightly different from our previous discussions involving massive scalars and fermions; we are not integrating the gauge bosons out of the theory but instead are evaluating the 1PI effective action. Nevertheless, the manipulations are exactly the same since the one-loop contribution to the 1PI effective action is still a functional trace of the form Tr log(D 2 + U ).
In evaluating the 1PI effective action, we split the gauge boson into a background piece plus fluctuations around this background, A µ = A B,µ +Q µ , and perform the path integral over the fluctuations Q µ while holding the background A B,µ fixed. In order to do the path integral, one must gauge fix the Q µ fields. At first glance, one might think that gauge fixing destroys the possibility of keeping gauge invariance manifest while evaluating the one-loop effective action. However, this turns out not to be the case. It is well known that there is a convenient gauge fixing condition that leaves the gauge symmetry of the background A B,µ field manifest, i.e. it only gauge fixes Q µ and not A B,µ . This technique is known as the background field method (for example, see [58] and references therein). 51 Because the gauge symmetry of the background A B,µ field is not fixed, we will still be able to employ the techniques of the covariant derivative expansion, allowing a manifestly gauge invariant computation of the one-loop effective action.
The issues around gauge symmetry are actually quite distinct for the background field method versus the CDE. However, because similar words are used in both discussions, it is worth clarifying what aspects of gauge symmetry are handled in each case. The background field method makes it manifestly clear that the effective action of A B,µ possesses a gauge symmetry by only gauge fixing the fluctuating field Q µ . This is an all orders statement. However, when evaluating the effective action order-by-order, one still works with the non-covariant quantities A B,µ , Q µ , and ∂/∂x µ at intermediate steps. 52 The covariant derivative expansion, on the other hand, is a technique for evaluating the oneloop effective action that keeps gauge invariance manifest at all stages of the computation by working with gauge covariant quantities such as D µ . To understand this point more explicitly, one can compare the method of the CDE presented in this paper and in [15] with the evaluation of the functional determinant using the component fields as presented in detail in Peskin and Schroeder [59]. Now onto the calculation. We take pure SU (N ) gauge theory, where F µν = F a µν t a and we take the t a in the adjoint representation, tr t a t b = N δ ab , (t b ) ac = if abc .We denote the covariant derivative as D µ = ∂ µ − iA µ with the field strength defined as usual, Note that we have normalized the gauge field such that the coupling constant does not appear in the covariant derivative. 51 All techniques of evaluating effective actions are, by the definition of holding fields fixed while doing a path integral, background field methods. Nevertheless, the term "background field method" is usually taken to refer to employing this special gauge fixing condition while evaluating the 1PI effective action. 52 To one-loop order, one only deals with AB,µ and ∂µ.
Let Γ[A B ] be the 1PI effective action. To find Γ[A B ], we split the gauge field into a background piece and a fluctuating piece, A µ = A B,µ + Q µ , and integrate out the Q µ fields. 53 The one-loop contribution to Γ comes from the quadratic terms in Q µ . We have Note that D µ = ∂ µ − iA B,µ and G µν = i[D µ , D ν ] are the covariant derivative and field strength of the background field alone. In order to get sensible results out of the path integral, we need to gauge fix. As in the background field method, we employ a gauge fixing condition which is covariant with respect to the background field A B,µ . Namely, the gauge-fixing condition G a is taken to be G a = D µ Q a µ . The resultant gaugefixed Lagrangian-including ghosts to implement the Fadeev-Popov determinant-is, e.g. [58,59], where ξ is the gauge-fixing parameter. The utility of this gauge fixing condition is that the fluctuating Q a µ is gauge fixed while the Lagrangian (A.9c) together with L g.f. + L gh posseses a manifest gauge symmetry with gauge field A B,µ that is not gauge fixed. Thus we can perform the path integral over Q a µ while leaving the gauge invariance of the effective action of A B,µ manifest. Under a background gauge symmetry transformation, A B,µ transforms as a gauge field, A B,µ → V (A B,µ + i∂ µ )V † while Q µ (and the ghosts c and c) transforms simply as a field in the adjoint representation, Q µ → V Q µ V † . Procedurally, when performing the path integral over Q and c, one can simply think about these fields as regular scalar and fermion 54 fields in the adjoint of some gauge symmetry and with interactions dictated by the Lagrangians in (A.9c) and (A.10).
The quadratic piece of the combined Yang-Mills, gauge-fixing, and ghost Lagrangian is We will work in Feynman gauge with ξ = 1 so that we can drop the D µ D ν term. Note that everything inside the square brackets in the above is in the adjoint representation (recall, f abc = −i(t b ) ac ). Using the generator for Lorentz transformations on four-vectors, (J ρσ ) µν = i(δ µ ρ δ ν σ − δ ν ρ δ µ σ ), we can write 53 To keep our discussion short, we are being slightly loose here. In particular, a source term J for the fluctuating fields needs to be introduced. After integrating out the fluctuating field, we obtain an effective action which is a functional of J and the background fields, W [J, AB]. The 1PI effective action, Γ[AB], is obtained by a Legendre transform of W . For more details see, for example, [58]. 54 Of course ghosts aren't fermions; they are anti-commuting scalars. We are speaking very loosely and by fermion we are referring to their anti-commuting properties.
The quadratic piece of the Lagrangian is then given by where 1 4 is the 4 × 4 identity matrix for the Lorentz indices, i.e. (1 4 ) µ ν = δ µ ν . Performing the path-integral over the gauge and ghost fields we obtain where the factor of 1/2 in the first term is because the Q a µ are real bosons, while the factor of −1 in the second term is because the c a are anti-commuting. Note that the functional traces makes totally transparent the role of the ghosts. The trace of the gauge boson term containing D 2 picks up a factor of 4 from the trace over Lorentz indices, one for each Q µ µ = 0, 1, 2, 3. Of course, the gauge boson only has two physical degrees of freedom; we see explicitly above that the ghost piece cancels the contribution of two of the degrees of freedom.
Each of the traces in the above are of the form Tr(−P 2 + U ), and thus we can immediately apply the transformations leading to the covariant derivative expansion. Switching to our notation G ′ µν = [D µ , D ν ] = −iG µν and defining we have where G and U gauge are defined as in Eq. (2.22) with U → U ferm . The first term in the above is from the fluctuating gauge fields, while the second is from the ghosts. Note also that the trace "tr" in the first term includes over the Lorentz indices, just as the trace for fermions in Eq. (A.7) is over the Lorentz (spinor) indices. In fact, it should be clear that U gauge is very similar to the first term in U ferm (Eq. (A.6)): U ferm ⊃ −i(σ µν /2)G ′ µν where σ µν /2 is the generator for Lorentz transformations on spinors.
Note that the effective action (A.15) contains infrared divergences from the massless gauge and ghost fields that we integrated out. These divergences can be regulated by adding a mass term for Q a µ and c a because these mass terms respect the gauge invariance of the background field A B,µ . 55

Massive gauge bosons
With our understanding of the story for massless gauge bosons, it turns out to be simple to obtain the result for massive gauge bosons. We consider massive vector bosons Q µ transforming under an unbroken, low-energy gauge group. As is well known, beyond tree-level perturbation theory, the Nambu-Goldstone bosons (NGBs) "eaten" by the massive vector boson must be included, i.e. we cannot work in unitary gauge. By working in a generalized R ξ gauge, we will be able to maintain manifest covariance of the low-energy gauge group. As we will see, mathematically, the results are essentially the same as the the massless case in Eqns. ((A.12),(A.13)), modified by the presence of mass terms for the Q µ and ghosts as well as an additional term for the NGBs.
Assuming that there is a weakly coupled renormalizable UV model, 56,57,58 we consider a general picture that the full gauge symmetry group G of the UV model is spontaneously broken into a subgroup H. A set of gauge bosons Q i µ have "eaten" the Nambu-Goldstone bosons χ i and obtained mass m Q due to Higgs mechanism. We assume this symmetry breaking is done by an extremely heavy "Higgs" with mass m H ≫ m Q , and hence the physical degree of freedom of which is safely decoupled. Let G have a general structure (A. 16) Let T A be the set of generators of G, with A = 1, 2, . . . , dim(G). Due to Eq. (A. 16), the set of generators T A are composed by a number of subsets denote the structure constant of G : where G A µ denote the gauge fields, and G A µν the field strengths, and g A are the gauge couplings that could be arbitrarily different for T A of different subsets in Eq. (A.17). Here we emphasis that the above expression of the full covariant derivative holds for any representation of G. 56 In general, this need not be the case. For example, the Qµ could be composite particles in the low-energy effective description of some strongly interacting theory. Another example is when additional massive vector bosons are needed to UV complete the theory. For example, an effective theory with a massive vector transforming as a doublet under a SU (2) gauge symmetry is non-renormalizable-a valid UV completion could be an SU (3) gauge symmetry broken to SU (2), but this requires an additional doublet and singlet vector. For our present purposes, relaxing this assumption only alters one result, which we discuss below. 57 As in all the other cases considered in this work, although never explicitly stated, we are also assuming the fields we integrate out are weakly coupled amongst themselves and the low-energy fields, so that it makes sense to integrate them out. 58 G itself may be contained in some larger group G which also contains exact and approximate global symmetries and the same mechanism responsible for breaking G → H may also break some of these global symmetries. These generalities do not affect our results below, which concern the transformation of Qµ and its associated NGBs under H. We therefore stick to our simplified picture for clarity.
Because we have put the arbitrary gauge couplings into the covariant derivative, the gauge boson kinetic term of the UV Lagrangian is simply In order to write this kinetic term in terms of the full covariant derivativeD µ , let us define an inner product in the generator space {T A }: which just looks like a scaled version of trace. However, we emphasize that, although it should be quite clear from definition, this inner product is essentially very different from the trace. The inner product can only be taken over two vectors in the generator space, while a trace action can be taken over arbitrary powers of generators. Nevertheless, the inner product defined in Eq. (A.22) has many similar properties as the trace action. For example, if one of the two vectors is given in a form of a commutator of other two generators, a cyclic permutation is allowed Note that the second line above is true because for the case g A = g C , f ABC = 0. As we shall see shortly, this cyclic permutation property will play a very important role in our derivation. With the inner product defined in Eq. (A.22), the gauge boson kinetic term Eq. (A.21) can be very conveniently written as Now let us consider the subgroup H of G. Let t a be the generators of H, which span a subspace of the full group generator space, and have closed algebra with f abc H denotes the structure constant of H, and a = 1, 2, ..., dim(H). Once the full group G is spontaneously broken into H, it is obviously convenient to divide the full generator space into the unbroken generators t a and the broken generators X i , i = 1, 2, ..., dim(G) − dim(H), with the corresponding massless gauge fields A a µ and massive gauge bosons Q i With all the above preparations, we are eventually ready to decompose the full gauge boson kinetic term in Eq. (A.21). First, the commutator of the full covariant derivative is Keeping only terms relevant and up to quadratic power for Q i µ , it follows from Eq. (A.24) that After adding up the gauge fixing and ghost piece in a general R ξ gauge, the total Lagrangian up to quadratic terms in Q i µ is We would like to stress that in spite of the allowance of arbitrary gauge couplings for each simple group G i , the end kinetic term of the heavy vector boson Q i µ has the above universal form. Another way of understand this is from the tree-level unitarity. Let us consider one component of the massless background gauge boson and call it a "photon" A µ with its coupling constant e and generator Q. The first two terms are obviously the kinetic term for the heavy vector boson Q a µ coupled to photon according to its charge q a , because Here, we used a complex linear combination of generators such that it is an "eigenstate" of the charge, [Q, T a ] = q a T a . The "curl" form derives from the original Yang-Mills Lagrangian in the UV theory. What is the least obvious is the universal coefficient for the "magnetic dipole term" tr[Q µ , Q ν ]W µν . Focusing on the "photon" coupling, it is The operator is gauge invariant under the unbroken gauge symmetry and one may wonder whether the coefficient can be arbitrary and model dependent.
It is well known that the amplitude for γγ → W + W − would grow as E 2 W in the Standard Model if the magnetic dipole moment κ = 1. The quadratic part of the Lagrangian above is sufficient to determine the tree-level amplitude, and the diagrams are exactly the same as those in the Standard Model. Unless κ = 1, it violates perturbative unitarity at high energies. Because the amplitude does not involve the Higgs or other heavy vector bosons, the amplitude is exactly the same as that in the UV theory, which is unitary. Therefore, the perturbative unitarity for this amplitude needs to be satisfied with the quadratic Lagrangian, which requires the dipole moment to have this value.

A.2 Useful identities
Commutators/anti-commutators 60 And hence we have Derivatives and integrals 60 Note that we are not distinguishing upper and lower indices, so in the following, δµν here is should be understood as gµν.
The following are useful integrals. They are in Minkowski space, and the powers of the free propagator-n in ∆ n -is assumed large enough to make the integral converge:

Operator identities and trace computations
Let us state some basics of covariant derivative calculus. Most of these are obvious, but we list them here because we make use of them over and over in calculations.
• The covariant derivative acting on a matrix is given by the commutator, D µ A = [D µ , A].
• The basic rules of calculus are the same. In particular, the chain rule holds: D(AB) = (DA)B + A(DB). This implies integration by parts holds, dxtr A(DB) = dxtr − (DA)B .
A term that often shows up in calculations is Breaking I n into easier to work with pieces, we define integrals involving only G as J n and integrals involving only U as K n , J n ≡ dq dm 2 tr ∆ − {q G} − G 2 n ∆, We define L n for integrals involving mixed G and U terms as L n ; for example, L 2 is given by · tr A 5 , · tr A 6 .

B More on mapping c i to physical observables
This appendix shows the calculational details of the mapping step described in our Section 4. We first list out in Appendix B.1, all the relevant two-point and three-point Feynman rules from the set of dimension-six operator in Table 4. Transverse vacuum polarization functions, that can be readily read off from the two-point Feynman rules, are also tabulated. Then in Appendix B.2 and B.3 we present details in calculating the "interference correction" ǫ I for Higgs decay widths and Higgs production cross sections respectively. We list out relevant Feynman diagrams, definitions of auxiliary functions, and conventional form factors. Finally, in Appendix B.4 and B.5, we show our calculation steps of the residue modifications and the parameter modifications, which are related to the "residue correction" ǫ R and the "parametric correction" ǫ P respectively.

B.1.1 Feynman rules for vacuum polarization functions
Throughout the calculations in the paper, the relevant vacuum polarization functions are those of the vector bosons iΠ µν V V (p 2 ) ∈ iΠ µν W W (p 2 ), iΠ µν ZZ (p 2 ), iΠ µν γγ (p 2 ), iΠ µν γZ (p 2 ) and that of the Higgs boson −iΣ(p 2 ). It is straightforward to expand out the dim-6 effective operators listed in Table 4, identify the relevant Lagrangian pieces, and obtain the Feynman rules. The relevant Lagrangian pieces are shown in Eq. (B.5)-Eq. (B.9). The resulting Feynman rules of the vacuum polarization functions are drawn in Fig. 4, with the detailed values listed in Eq. (B.10)-Eq. (B.14). In the diagrams, we two sets where we have adopted the notation c Z ≡ cos θ Z etc., with θ Z denoting the weak mixing angle. This alternative set of vector boson transverse vacuum polarization functions are summarized in Table 13.

B.2 Details on interference corrections to the Higgs decay widths
There is no new amputated diagrams for h → ff decay modes up to leading order (linear power and tree level) in Wilson coefficients, because we are considering only the bosonic dim-6 effective operators ( Table 4). The h → gg, h → γγ, and h → γZ decay widths are already at one-loop order in the SM, so the only new amputated diagram up to leading order in Wilson coefficients is given by the new three-point vertices iM µν hgg (p 1 , p 2 ), iM µν hγγ (p 1 , p 2 ), and iM µν hγZ (p 1 , p 2 ) ( The h → W W * and h → ZZ * modes are a little more complicated, because they are at tree level in the SM. It turns out that there are two new amputated diagrams for h → W W * mode as shown in Fig. 6, and four new amputated diagrams for h → ZZ * mode as shown in Fig. 7. It is straightforward to evaluate these relevant new diagrams using the new Feynman rules listed in Section B.1 (together with the SM Feynman rules). One can then compute the interference correction ǫ I for each decay mode from its definition (Eq. (4.11)). The three-body phase space integrals are  analytically manageable, albeit a little bit tedious. We summarize the final results of ǫ I in Table 8, The A SM hgg , A SM hγγ , and A SM hγZ in Table 8 are the standard form factors , and A 1/2 (τ ), A 1 (τ ), A 1/2 (τ, λ), A 1 (τ, λ) being the conventional form factors (for example see [60]) A 1/2 (τ ) = 2τ −2 τ + (τ − 1) f (τ ) , (B.36) A 1 (τ ) = −τ −2 2τ 2 + 3τ + 3 (2τ − 1) f (τ ) , (B.37)

B.3 Details on interference corrections to Higgs production cross section
The ggF Higgs production mode is just the time reversal of the h → gg decay. Again as it is already at one-loop order in the SM, the only new amputated diagram up to leading order in Wilson coefficients is given by the new three-point vertex iM µν hgg (p 1 , p 2 ) (Fig. 5(e)) multiplied by the polarization vectors iM ggF, AD,new = iM µν hgg (p 1 , p 2 )ǫ µ (p 1 )ǫ ν (p 2 ).

(B.42)
Obviously, the interference correction to ggF production cross section is the same as that to h → gg decay width The vector boson fusion production mode σ W W h has three new amputated diagrams as shown in Fig. 10 (in which one of the fermion lines can be inverted to take account of production mode in lepton colliders such as the ILC). For the vector boson associate production modes, there are two new diagrams for σ W h (Fig. 8) and four for σ Zh (Fig. 9). Again from the definition (Eq. (4.11)), we compute the interference correction ǫ I for each Higgs production mode. The final results are summarized in Table 10. For σ W h and σ Zh , the final states phase space integral is only two-body and quite simple. On the other hand, σ W W h requires to integrate over a three-body phase space, which turns out to be quite involved. The analytical result ǫ W W h,I (s) is several pages long and hence would not be that useful. Instead, we provide numerical results of it in Table 10, where three auxiliary functions f a (s), f b (s), and f c (s) are defined. We provide the numerical results of these auxiliary functions (Fig.3) as well as mathematica code of their calculations.
To show the definition of f a (s), f b (s), and f c (s), we need to describe the three-body phase space integral of σ W W h . We take the center of mass frame of the colliding fermions and setup the spherical coordinates with the positive z-axis being the direction of

B.5 Calculation of Lagrangian parameter modifications
The set of Lagrangian parameters relevant for us are {ρ} = {g 2 , v 2 , s 2 Z , y 2 f }. We would like to compute them in terms of the physical observables and the Wilson coefficients ρ = ρ(obs, c i ), where the set of observables relevant to us can be taken as {obs} = {α,Ĝ F ,m 2 Z ,m 2 f }. We put a hat on the quantities to denote that it is a physical observable measured from the experiments. On the other hand, for notation convenience, we also define the following auxiliary Lagrangian parameters that are related to the basic ones {ρ} = {g 2 , v 2 , s 2 Z , y 2 f }: (B.59) These auxiliary Lagrangian parameters are not hatted.
As explained in Section 4, in order to obtain ρ = ρ(obs, c i ), we first need to compute the function