Functional Matching and Renormalization Group Equations at Two-Loop Order

We present a systematic method for determining the two-loop effective Lagrangian resulting from integrating out a set of heavy particles in an ultraviolet scalar theory. We prove that the matching coefficients are entirely determined from the (double-)hard region of the loop integrals and present a master formula for matching, applicable to both diagrammatic and functional approaches. We further employ functional methods to determine compact expressions for the effective Lagrangian that do not rely on any previous knowledge of its structure or symmetries. The same methods are also applicable to the computation of renormalization group equations. We demonstrate the application of the functional approach by computing the two-loop matching coefficients and renormalization group equations in a scalar toy model.


Introduction
The use of Effective Field Theories (EFTs) in beyond the Standard Model (BSM) searches is ubiquitous.Given the current experimental bounds, it seems increasingly likely that there is a large gap between the electroweak scale and the next energy threshold.In this event, EFTs are the ideal tool to capture the possible low-energy effects of new physics (NP), whatever it might be.With them, we can look for subtle NP effects, opening the door to the exploration of energy scales orders of magnitude beyond what can be reached on-shell at the LHC.
It is now a decade since the computation of the oneloop RG equations in the Standard Model Effective Theory (SMEFT) [33][34][35], and we wonder if one-loop corrections are sufficient for the current precision require- * javier.fuentes@ugr.es† ajdin.palavric@unibas.ch‡ anders.thomsen@unibe.chments.Indeed, certain low-energy effects are generated only at two-loop order [36].Furthermore, both the strong and the top Yukawa couplings are large enough that they might generate considerable running contributions, such as in [37,38].On top of this, the inclusion of two-loop RG effects becomes mandatory if one wants to restore the scheme independence in one-loop matching calculations [39,40], making them an important ingredient in the automated one-loop matching endeavor.With this letter, we take the first step towards efficient, functional two-loop RG and matching calculations.It should come as no surprise that such a step is possible, as other variations of functional methods have been used for the calculation of the quantum effective potential [41,42] and the counterterms of chiral perturbation theory [43] with heat-kernel methods.Both of these calculations have also been performed at two-loop order [44,45].What is perhaps more remarkable is that the functional formalism lets us prove a generic master formula for two-loop matching, where we directly identify the two-loop EFT action with the hard part of the ultraviolet (UV) quantum effective action.With this formula, there is no need to identify cancellations between EFT and UV contributions on a case-by-case basis.It also opens up the future possibility of automating (functional) matching beyond one-loop order.
In this letter, we outline the extension of functional methods to two-loop order in the simpler case of scalar theories.In Section 2, we present our functional method, with a proof to the generic matching formula supplied in Appendix A. Next, in Section 3, we demonstrate the practical application of the method by calculating the two-loop matching of a scalar toy model to its low-energy EFT and the two-loop RG equations of the resulting theory.We leave further details of the method along with extensions to fermionic degrees of freedom and gauge theories to a forthcoming paper.
The vacuum functional, W[J], is the generating functional of all connected Green's functions of a theory and, therefore, contains all physical information.If we take the fields (including their conjugates) to be collectively denoted by η a (x) and the action by S[η], the vacuum functional is defined by the path integral: where the subindices with capital Latin letters I, J, . . .correspond to DeWitt notation, where the spacetime dependence is included as part of the label, e.g.I = (x, a).Thus, the contraction of repeated indices denotes not only an implicit summation in the field labels but also an integration over spacetime. 1 The path integral can be evaluated perturbatively, using a saddle-point approximation around the classical background-field configuration η, which is the solution to the tree-level equations of motion (EOMs), that is where the superindex denotes loop order, with (0) indicating tree level.We parameterize the expansion of the action around the background-field configuration as IJ .For renormalizable theories, all higher-loop vertices A (ℓ) IJ , . . .for ℓ ≥ 1 stem from the counterterms of the renormalized action; however, this could be different if our UV theory is itself an EFT.
Using the defining relation of the vacuum functional given by (1) together with the expansion of the action around the background field, the vacuum functional can be written in the form The expansion of the vacuum functional reproduces the usual loop expansion (formally in ℏ).Therefore, the terms of O(ℏ 2 ) in (4) correspond to the two-loop topologies, while the terms of higher order are dropped.
It is convenient to relate this functional to the quantum effective action, Γ, as we will ultimately relate Γ to the EFT action.The quantum effective action is the generating functional of all one-particle-irreducible (1PI) Green's functions of the theory and is defined by the Legendre transform of W: The background field η and the sources then satisfy the quantum EOM 1 In the case at hand, J I η I = x Ja(x)ηa(x) with x ≡ d d x.
Plugging this back into definition (5) along with the saddle-point approximation of W, we obtain where, analogously to the bar notation, the hat is used as shorthand for exclusive dependence on η, e.g.
. This is an extension of the well-known expression for the effective action at one-loop order, where all one-loop contributions are contained in a functional (super)trace.Interestingly, the terms in the expression above can be understood as vacuum graphs with Q −1 acting as a quantum-field propagator, dressed with arbitrary insertions of the background fields η, and B (1) , C (0) and D (0) corresponding to two-, three-, and fourpoint quantum field interactions (again in the presence 1. Graphical representation of the O(ℏ2 ) terms appearing in the effective action in (7).
of background fields).This is represented in Figure 1.
Each closed loop in these dressed graphs can be associated to an integration over a loop momentum, in a similar manner to traditional Feynman graphs.
In general, evaluating the effective action ( 7) can be very complicated because of the need for inverting Q and evaluating C at different spacetime points.Nonetheless, when all loop momenta are restricted to a hard region, 2 these quantities can be evaluated directly in terms of an operator-product expansion around the hard scale.As we will now discuss, the hard-momenta region of Γ is all that is needed for EFT matching and RG evolution, also at two-loop order.

Master formula for EFT matching
Let us consider a weakly-coupled UV theory, S UV [η], where η I = (Φ α , ϕ i ) denotes the collection of all fields, heavy and light, respectively.We seek to determine an EFT action S EFT [ϕ] that reproduces the physics of the full theory at energies much below the masses of the heavy fields, which we assume to lie around a generic scale Λ.
In off-shell matching computations, we begin with an even stronger requirement for the EFT matching condition: the EFT should reproduce all low-energy Green's functions of the full theory.This stronger requirement lets us consider the generating functional of the theories rather than the S-matrix.Thus, our aim is to determine That is, we enforce equality of all connected Green's functions involving the light fields.Each side of ( 8) is to be understood in terms of power series in 1/Λ.To make matters simpler, we proceed with a Legendre transformation of the light-field sources in order to frame the matching condition in terms of the quantum effective actions: The heavy fields Φ[ φ] are solutions to the quantum EOMs in the presence of the light background fields.In diagrammatic terms, condition ( 9) equates all one-lightparticle-irreducible (1LPI) Green's functions of the UV theory with the 1PI Green's functions of the EFT.It was demonstrated in [12,13] that, at one-loop order, there is a cancellation between the loop contributions in the EFT and the soft-region of the loops in the UV theory.This enables a very direct computation of S EFT [ϕ] in terms of the hard region of the UV quantum effective action.
We are now ready to generalize these considerations and present a master formula for perturbative EFT matching at multi-loop order: the off-shell EFT action is determined by Here the 'hard' part is taken to include all contributions without any soft loops.It includes tree-level contributions as well as loops where all loop momenta are hard.In many ways, this is an intuitive leap: the hard, local part of the UV theory is identified with the EFT action, while the long-distance physics is captured by loops in the EFT.Nevertheless, we have never seen an explicit formulation of this notion, much less a matching formula applicable to practical computations.A constructive proof of the matching formula (10) at two-loop order is provided in Appendix A, where we show that there is a one-to-one correspondence between (partially) soft loops in the UV and loops in the EFT.We postpone the discussion on a possible extension of this proof to higher-loop orders to a more comprehensive follow-up paper.The end result is that all loop integrals needed to perform the EFT matching (i.e.those in the hard limit) reduce to vacuum integrals, for which expressions are known up to three loops [48].
The matching formula ( 10) is a d-dimensional off-shell relation.A complication associated to this kind of matching is that it does not produce the EFT Lagrangian directly in a four-dimensional on-shell basis.Rather, one has to apply field redefinitions to reduce the output to an on-shell basis.Likewise, the matching result may also produce EFT operators that are not present in a fourdimensional basis.As a result, one has to separate out evanescent operators and, preferably, convert the EFT action to an evanescence-free scheme [49][50][51][52][53][54][55].A related consideration to be aware of beyond one-loop order is that lower-order matching coefficients may contain O(ϵ) contributions.These cannot be ignored, as their insertion in an EFT loop can lead to finite contributions.Therefore, one has to carefully remove the O(ϵ) terms and absorb them into finite coefficients at higher-loop order, similarly to what is done for evanescent contributions [52].

Functional approach for RG evolution
The MS (or MS) counterterms of a theory, S[η], can also be determined from the effective action from the observation that it must be free of UV divergences.If we use dimensional regularization to regularize the loop integrals, finiteness of the renormalized generating functional translates to the condition where K ϵ is an operator that extracts all 1/ϵ poles of UV origin.This equation establishes a relation between the MS counterterms of the theory, identified with the UV poles of S (ℓ) , and the other terms in the effective action.Denoting the MS counterterms by δ S which, together with the expression of Γ and condition (11), establishes a direct relation to determine δ S (ℓ) MS functionally.Other renormalization schemes of the subtraction family can also be obtained by appropriately adapting the definitions of the counterterms in the expression above.
Restricting to the two-loop expression of the effective action in (7), we obtain the following MS counterterms up to two-loop order: A convenient prescription for extracting the UV poles in theories with massless states consists in introducing a common mass, Λ, in all propagators of the loop integrals.This mass acts as a hard scale (assumed to be much larger than any other scales in the loop integrals) and serves as an infrared regulator.The overall UV divergence of a loop integral is identified with the part where all loop momenta are large.Hence, it is easy to show that the UV divergences can be extracted from the hard-region (defined by all loop momenta being of order Λ) of the loop integrals [56].This effectively establishes a powercounting on Λ, around which Q −1 can be expanded.As in the matching case, the resulting loop integrals are just vacuum integrals (in this case with a single mass Λ) for which results are known up to three loops [48].The only drawback of this method for UV-pole extraction is that one also needs to consider spurious counterterms with positive powers of Λ.These spurious counterterms can break the symmetries of the original Lagrangian and are needed for the cancellation of subdivergences.After the counterterms have been determined, the RG equations can be readily obtained via standard techniques.

Functional evaluation of the effective action
The functional evaluation of the effective action requires manipulating the inverse dressed propagator, Q, which can be generically parametrized by with P µ x = i∂ µ x denoting the momentum operator.Since the action is local, it is always possible to factor out a delta function.This is also the case when dealing with more complicated functional forms involving Q, such as ln Q or Q −1 .It is well-known that the one-loop contribution to the quantum effective action, c.f. (7), is given by the functional (super)trace This expression is a nonlocal function of Q and is difficult to evaluate in general.However, as we are only interested in the hard region, where the loop momentum is taken to be of the order of the heavy scales, we can perform an operator-product expansion.For scalar theories, the inverse dressed propagator takes the generic form with M a being a possible (hard) mass, and U ab a generic interaction term which, as we make explicit in its argument, may involve derivatives.Denoting the operator-product expansion of the logarithm reads where (ln ∆ −1 ) ab = δ ab ln(k 2 − M 2 a ) contributes to an unphysical constant that is subtracted when normalizing the path integral and the expansion for ∆ takes the form Likewise, the dressed propagator, necessary for evaluating the two-loop contributions, admits the expansion The hard-region evaluation guarantees that subsequent terms in all these series are further suppressed, so only a finite number of terms need to be retained to a given order in the EFT expansion.A manifestly local result is obtained only in the hard-region limit.
Locality of the action likewise ensures that delta functions can be factored out of the remaining functional objects.We write the quantum-field interactions as series in the momenta operator: 3

B
(1) In most practical applications, only a small number of terms from this momentum operator expansions are present.In fact, for renormalizable theories, only the terms with at most two momenta in B, one in C, and none in D are nonzero.
Having made these definitions, we can now evaluate the two-loop contributions to the effective action in (7).We parameterize them as where the loop contributions G i are identified with the counterterm, figure-8, and sunset topologies, respectively (as depicted in Figure 1).The main subtlety in the functional evaluation of the two-loop effective action is related to the sunset topology, where it is necessary to power expand one of the two vertices around the location of the 3 For compactness, we employ a power-like notation with underlined superscripts for the Lorentz indices, that is, we denote B (1) m ab P m x ≡ B (1) µ 1 ...µm ab other.We then obtain integral formulas for the functional contractions and, using a momentum-space representation for the delta functions, we find the expressions In the equations above, any P x acting to the rightmost of a bracket yields a null contribution.This would not be the case anymore in the gauge non-singlet scenario, where the derivative in P x would be promoted to a covariant derivative.A manifestly covariant generalization of these expressions will be presented in a follow-up paper.While the sum in s in the last expression runs to infinity, only a finite number of terms need to be retained at a given order in the EFT counting when considering the hardmomentum limit.In particular, only terms up to s = 4 contributes at EFT dimension six.

A toy-model example
We illustrate the functional method described in the previous section with a concrete example: a toy-model, consisting of one heavy and one light real scalar fields, Φ and ϕ, respectively.For simplicity, we assume that the theory possesses a Z ϕ 2 × Z Φ 2 symmetry, with Z Φ 2 softly-broken by a trilinear term with coupling κ ≪ M .This soft-breaking term is included to allow for a non-trivial quantum EOM of the heavy-field, cf.(10).The UV Lagrangian of this theory is given by with the corresponding UV counterterms L ct UV defined as where we anticipate that new Z Φ 2 -breaking interactions are generated radiatively and need to be renormalized. 4

One-loop UV counterterms
Our calculations are done in a tadpole-free MS scheme (for both UV theory and the EFT), such that the Φ tadpole is removed with a finite counterterm.Given this scheme choice, the determination of the finite parts of the two-loop matching conditions requires only the calculation of the one-loop counterterms.From ( 13) and with the functional evaluation described in Section 2.4, it follows that5 L ct ( 1) where a = Φ, ϕ.The functional objects relevant for this calculation are given by and the expansion of ∆ in (19).As the UV theory in our example contains no massless states, no IR divergences appear and the UV divergences are readily obtained.
Only the terms with n ≤ 2 in the sum above contribute to the UV divergences.Denoting i + O(ℏ 2 ), we find where ln M 2 ≡ ln M 2 /μ 2 and μ is the MS renormalization scale.The value of these counterterms has been cross-checked against the RG functions obtained with RGBeta [57].

EFT matching at two-loop order
For the present example, it is easy to see that there is no tree-level contribution to the EFT action, as the Lagrangian has no linear dependence on the heavy field Φ.The one-loop part of the EFT Lagrangian is calculated as usual from the hard region of the functional (super)trace which, after doing the expansion (18), yields L with ∆ and X provided in ( 19) and ( 27), respectively.
In contrast with the counterterm evaluation, we are now interested in the hard-momentum region defined by the relation k ≳ M ≫ m ϕ , with k being the loop momentum.
In practice, this implies that, for this calculation, the mass term in ∆ −1 ϕϕ needs to be power-expanded before loop integration.That is, One difference to keep in mind with respect to the usual one-loop matching evaluations is the need to retain O(ϵ) terms.As described in Section 2.2, these terms can be shifted into the two-loop matching coefficients with an appropriate modification of the renormalization conditions.The one-loop EFT Lagrangian resulting from the functional trace is a function of both heavy and light fields.The former are removed by means of the heavyfield EOMs defined in (10).In our case, we find that The contributions from replacing Φ by its EOM yield two-loop effects when inserted back into the one-loop (and tree-level) EFT Lagrangian.No terms of two-loop order are needed in our toy-model calculation, as these would start contributing only at the three-loop level.The two-loop EFT Lagrangian follows directly from the evaluation of expressions (23) in the double-hard region, defined by k, ℓ ≳ M ≫ m.In this example, only terms without powers of P x,y,z appear in B (1) , C, and D, and thus, only the terms with m (′) = n (′) = r = 0 in (23) contribute.We have where we dropped terms containing Φ, as these contribute only at higher loop orders.The final (off-shell) EFT Lagrangian is obtained by combining the results from (23) with the contributions from the Φ-field EOM and the two-loop shift from removing the one-loop O(ϵ) terms.The resulting EFT Lagrangian can be written in terms of an on-shell operator basis via appropriate ϕ-field redefinitions.For concreteness, we present here this final on-shell result: with the two-loop matching conditions between the EFT and UV Lagrangians taking the form where Cl 2 (x) is the Clausen function of order 2 with Cl 2 ≡ Cl 2 (2π/3) ≈ 0.6766277.To our knowledge, this is the first two-loop matching computation performed with functional methods.As a crosscheck of our result, we verified that all single-and double-logarithmic contributions are consistent with the RG functions in both the UV theory 6 and the EFT such as to ensure matching scale independence.The determination of the RG functions in the EFT using functional methods is discussed in the next section.

Renormalization group functions at two-loop order
The EFT counterterms are calculated functionally using expressions (13).Analogously to the UV counterterms, the EFT counterterms at one-loop order are given by 6 The two-loop RG functions in the UV theory necessary for this crosscheck can easily be determined using RGBeta [57].
with ∆ EFT as in (19) and X EFT being The counterterm Lagrangian from the expression above is obtained in an off-shelf basis, and can be reduced to an on-shell Lagrangian by appropriate field redefinitions.Parameterizing the on-shell EFT counterterms by and separating the couplings by loop order and power of the ϵ-pole as we find δ at one-loop order.The two-loop part of the counterterms is then obtained from the second line of (13).Once more, only terms with no powers of P x,y,z in B (1) EFT , C EFT , and D EFT are present, and we have that where we used that δ ϕ = 0 in this example.Inserting these operators into the expressions in (23) (where, again, only the terms with m (′) = n (′) = r = 0 contribute) together with the expansion of Q −1 in (20) and evaluating the loop integrals, we find the two-loop contributions to the counterterms.As before, those are obtained in an off-shell basis, which reduces to the on-shell result λ,1 = − c6,1 = − ϕ,2 = 0 , after appropriate field redefinitions.As a cross-check, we have verified that these counterterms satisfy the consistency conditions on the double-poles necessary for finite RG functions (see, e.g.[58]).
Having obtained the two-loop counterterms, we readily determine the anomalous dimension γ ϕ of the field ϕ along with the beta functions for m 2 , λ and c 6 couplings: Those terms of the beta functions that involve only renormalizable couplings have been cross-checked with RGBeta [57] while the contributions with the c 6 coupling are found in agreement with [59][60][61].

Conclusions and outlook
The discovery of physics beyond the SM is proving more challenging than initially anticipated.Given the precision increase associated with upcoming experimental searches and the absence of clear indications of the possible shape of NP, the use of novel and more precise EFT approaches becomes more important than ever.
In this letter, we have presented the initial steps toward functional EFT matching and RG evolution at two-loop order, so far restricted to the case of scalar theories.We have explicitly demonstrated that the hard part of the UV effective action is all that is needed for such calculations.To our knowledge, this is the first time that an explicit proof of this statement has been presented.While this result is applicable for both diagrammatic and functional approaches, it becomes particularly useful for the latter since it enables a power counting around which to perform an operator-product expansion of the functional results.Building on this, we have provided closed-form expressions for the evaluation of the two-loop effective action in the hard limit.In this way, and analogously to the one-loop functional result, our calculation of the twoloop EFT Lagrangian does not require the determination of the target EFT basis and essentially amounts to simple algebraic manipulations, making it particularly suitable for automation.We have also presented a toy-model example that illustrates the main rationale behind the application of our functional approach and, as a byproduct, we have verified recent literature results concerning the determination of two-loop RG equations in the geometric approach [61].
The extension of these methods to the more general case, including theories with fermions and/or gauge bosons, remains non-trivial and will be discussed in a forthcoming publication.In particular, we observe that the standard techniques to make the functional evaluations manifestly covariant [9,10] are no longer applicable at two-loop order, and new strategies are required.Likewise, the generalization of these results to higher-loop orders will also be explored in the future.

=
−J I , and the bar being shorthand for exclusive dependence on η, e.g. S ≡ S[η].It is convention to denote the inverse dressed propagator of the quantum field (also known as the fluctuation operator) by x − w)δ(y − w)δ(z − w) .(21)