One-loop Matching of the Type-III Seesaw Model onto the Standard Model Effective Field Theory

In previous works arXiv:2107.12133 and arXiv:2201.05082, we have performed the one-loop matching of both type-I and type-II seesaw models for neutrino masses onto the Standard Model Effective Field Theories (SMEFT). In the present paper, by matching the type-III seesaw model onto the SMEFT at the one-loop level, we complete this series of studies on the construction of low-energy effective field theories (EFTs) for the canonical seesaw models. After integrating out the heavy fermionic triplets in the type-III seesaw model via both functional and diagrammatic approaches, we find 33 dimension-six (dim-6) operators in the Warsaw basis and their Wilson coefficients, while the number of dim-6 operators is 31 (or 41) for the EFT of type-I (or type-II) seesaw model. Furthermore, we calculate the branching ratios of radiative decays of charged leptons in the EFT. Then, the relationship between the beta function of the quartic Higgs coupling $\lambda$ in the full theory and that of $\lambda^{}_{\rm EFT}$ in the EFT is clarified. Finally, we briefly discuss the phenomenological implications of three types of seesaw EFTs and propose working observables that are sensitive to the four-fermion operators, which could be used to distinguish among different seesaw models in collider experiments.


Introduction
In the past few decades, a large number of neutrino oscillation experiments have provided us with strong evidence that neutrinos are massive but extremely light, and the lepton flavor mixing is very significant [3]. To accommodate tiny neutrino masses, three types of canonical seesaw models have been proposed [4][5][6][7][8][9][10][11][12][13][14][15]. In these seesaw models, neutrinos turn out to be Majorana particles and the lightness of their masses can be ascribed to the largeness of heavy particle masses, which may be far beyond the electroweak scale Λ EW ∼ 100 GeV. In this case, it seems quite difficult to achieve the on-shell production of these heavy particles in current or even next-generation collider experiments. An alternative way to scrutinize the seesaw models of neutrino masses is to explore the off-shell effects of heavy states, which may lead to remarkable deviations of some observables from the predictions of the Standard Model (SM).
At this point, the SM effective field theory (SMEFT) is a well-established theoretical framework to interpret possible deviations from the SM predictions [16][17][18]. Respecting the SM gauge symmetries and particle content, the SMEFT supplements the SM with a series of higher-dimensional operators that are suppressed by the inverse power of the cutoff scale Λ. At the leading order of 1/Λ, there is a unique dimension-five (dim-5) operator, i.e., the Weinberg operator [19], which gives rise to tiny neutrino masses after the spontaneous gauge symmetry breaking. The discovery of neutrino oscillations can be regarded as the robust evidence for the Weinberg operator. In the literature, a lot of efforts have been made to find the minimal complete set of operators of various mass dimensions in the SMEFT [19][20][21][22][23][24][25][26]. Nevertheless, as the dimension of the operator becomes higher, the number of independent operators in the SMEFT will increase rapidly [27]. Fortunately, such operators are also strongly suppressed by higher powers of 1/Λ, rendering their impact on physical observables smaller. For this reason, the most commonly used basis is the Warsaw basis [16] of dim-6, including 59 independent baryon-number-conserving operators and 4 baryon-number-violating ones. In the near future, the experiments at the high-energy and highintensity frontiers will hopefully measure the Wilson coefficients of these dim-6 operators and thus offer useful information about the ultraviolet (UV) full theory, from which the relevant dim-6 operators arise.
On the other hand, provided a well-motivated and renormalizable UV theory, one can match it onto the SMEFT by integrating out heavy degrees of freedom above the cutoff scale Λ. In this way, the resultant Wilson coefficients of higher-dimensional operators are completely determined by the model parameters in the UV theory and thus highly correlated. Ref. [28] outlines the basic strategy to construct the EFT from a specific renormalizable UV theory. The tree-level contribution of any UV completion (with general scalar, spinor and vector fields and any types of interactions) to the Wilson coefficients of dim-6 operators in the SMEFT has been derived in Ref. [29]. At the loop level, many UV models as the extensions of SM are found to be able to generate those dim-6 operators via one-loop matching .
In view of the discovery of neutrino oscillations, the canonical seesaw models are indeed wellmotivated UV theories. The one-loop matching of type-I and type-II seesaw models onto the SMEFT has been carried out in previous works [1,2]. The EFT descriptions of the seesaw models are referred to as the seesaw effective field theories (SEFTs). In this paper, we continue to carry out the complete one-loop matching for the type-III seesaw model up to dim-6 effective operators. To this end, both functional and diagrammatic methods are implemented to integrate out the heavy fermionic triplets in the type-III seesaw model. As a cross-check, the results by using these two different methods are compared with each other. Several mistakes in Ref. [48], where the one-loop matching of the type-III seesaw model has been done via the functional approach, are corrected. The main motivation for such an investigation is two-fold. First, the fermionic triplets in the type-III seesaw model experience the SM gauge interactions and come with three flavors, so the one-loop construction of its low-energy EFT is more complicated than that for either type-I or type-II seesaw models. Second, only with all these types of EFTs for the seesaw models can one start to consider whether it is possible to find out physical observables at low energies to distinguish among three types of seesaw models. Further discussions about the SEFT-III are also given. We work in the one-loop SEFT-III and calculate the rates of lepton-flavor-violating rare decays of charged leptons, such as µ → eγ. It is demonstrated that the SEFT-III can exactly reproduce the results in the full theory in the large-mass limit. Then, the beta function of the quartic Higgs coupling λ EFT is discussed and related to that in the full theory. By doing so, we show that the one-loop matching plays an important role in reconstructing the beta function in the full theory from that in the EFT. Finally, based on the dim-6 four-fermion operators, a brief study on the possible way to distinguish among three types of seesaw models is presented.
The remaining part of this paper is structured as follows. In Sec. 2, we introduce the type-III seesaw model and establish our notations and conventions. Then, the tree-level matching up to dim-6 is accomplished by utilizing the equation of motion. The one-loop matching onto the SMEFT with functional and diagrammatic approaches is given in Sec. 3. The resulting dim-6 operators in the Warsaw basis and their Wilson coefficients are explicitly provided. In Sec. 4, we further discuss some phenomenological implications of the SEFTs. We summarize the main results and conclude in Sec. 5.

The Type-III Seesaw Model
The type-III seesaw model extends the SM by introducing three right-handed fermionic triplets Σ R in the adjoint representation of the SU(2) L gauge group and with the hypercharge Y = 0. The gauge-invariant Lagrangian for the full type-III seesaw model is written as [15] with the SM Lagrangian being where f = Q L , U R , D R , ℓ L , E R refer to the SM fermionic doublets and singlets, and H is the SM Higgs doublet with Y = 1/2 while H is defined as H ≡ iσ 2 H * . In addition, ℓ c L ≡ Cℓ L T and Σ c R ≡ CΣ R T are defined, where C ≡ iγ 2 γ 0 denotes the charge-conjugation matrix. Without loss of generality, we shall work in the mass basis of Σ R , namely, the mass matrix for Σ R is diagonal with It is worthwhile to note that three heavy states in the same fermionic triplet have the same mass, while three flavors of fermionic triplets have their own masses as indicated by their mass matrix. In Eq. (1), the covariant derivative reads For the fields in the fundamental representation of SU(2) L , we have T I = σ I /2 (for I = 1, 2, 3) with σ I being the Pauli matrices. For the adjoint representation, we take the representation matrices as (T I ) JK = −iϵ IJK (for I, J, K = 1, 2, 3), where ϵ IJK is the totally antisymmetric Levi-Civita tensor. As for the adjoint representation of the SU(3) c group, T A = λ A /2 is defined with λ A being the Gell-Mann matrices (for A = 1, 2, · · · , 8). In the subsequent discussions, we rewrite Σ R in the adjoint representation as Σ R ≡ σ I · Σ I R / √ 2 and introduce the three-vector in the weak-isospin space. Hence the Lagrangian in Eq. (1) can be recast into in which Σ = Σ R + Σ c R has been defined, and Tr ΣΣ c = Tr ΣΣ = Σ · Σ has been used. It is worth stressing that all quantities in boldface represent the vectors in the weak-isospin space, and their components are defined as In these notations, the dot in Eq. (3) is just the inner product between two three-vectors. With the Lagrangian in Eq. (3), it is straightforward to derive the equation of motion (EOM) for Σ, i.e., By solving the above EOM, one can find the classical field Σ c and then expand it with respect to 1/M Σ in order to obtain the local fieldΣ c . Integrating out the heavy field Σ at the tree level is equivalent to substituting the local classical fieldΣ c back into the Lagrangian of the UV theory.
Since we are interested in the operators in the SEFT-III up to dim-6, it is sufficient to retain the terms up to O(1/M 2 Σ ) in the local field, i.e., After inserting the local classical fieldΣ c in Eq. (5) into the full Lagrangian Eq. (3), we complete the tree-level matching of the type-III seesaw model onto the SMEFT with the dim-5 and dim-6 operators. Some comments on the results of this tree-level matching are in order.
• The tree-level Lagrangian of the SEFT-III turns out to be L tree SEFT-III = L SM − (1/2)(Ŷ Σ + Y c † Σ ) ·Σ c , whereΣ c is given in Eq. (5). Keeping the terms of O(1/M Σ ) in Eq. (5), we get the dim-5 Weinberg operator where the identity σ I ab σ I cd = 2δ ad δ bc − δ ab δ cd has been used and H † H = H † ϵH * = 0 is implied. One can see that the Weinberg operator O αβ 5 ≡ ℓ α L H H T ℓ βc L is successfully derived with the Wilson coefficient C • In a similar way, to obtain the dim-6 operators, one should take out the terms of order 1/M 2 Σ in Eq. (5). More explicitly, the dim-6 operator in the SEFT-III at the tree level reads which coincides the result in Ref. [51]. Converting the above dim-6 operator into those in the Warsaw basis, we get the following three operators Note that the lepton-flavor indices of these dim-6 operators and their Wilson coefficients have been suppressed.
Thus far we have matched the type-III seesaw model onto the SMEFT at the tree level. The unique dim-5 Weinberg operator and three dim-6 operators in the Warsaw basis, together with their Wilson coefficients in terms of the model parameters in the full theory, are derived. The technical details of the one-loop matching have been clearly explained and extensively applied to various UV theories in Refs. [35,42,43,[52][53][54][55][56]. Therefore, we shall just summarize the loop-level matching results in next section.

One-loop Matching
The publicly available packages Matchmakereft [55] and Matchete [56] have been designed to accomplish the one-loop matching of any renormalizable UV model onto the SMEFT, based on the diagrammatic approach and the functional approach, respectively. We have utilized these two packages to carry out the one-loop matching of the type-III seesaw model and made a careful cross-check to ensure the correctness of the matching results. In the following, we will list the complete one-loop matching results, including threshold corrections to the SM couplings, the matching conditions for the Weinberg operator and for the dim-6 operators in the Warsaw basis.

Threshold corrections
The renormalizable terms of dim-4 that already exist in the SM receive threshold corrections from the one-loop matching. We collect these corrections to the SM Lagrangian as below As the kinetic terms for the Higgs doublet, gauge bosons and fermions are modified, one has to normalize these fields to maintain the canonical form. For the Higgs doublet and fermions, we For the gauge bosons, one has to redefine the gauge-boson fields and gauge couplings g 1 and g 2 simultaneously, i.e., to keep the covariant derivative D µ intact. For notational simplicity, we introduce the effective parameter g eff and the threshold correction δg eff such that g eff = g + δg eff , where g is the original parameter belonging to All the threshold corrections are where "tr" indicates the trace operation in the flavor space, and with µ being the 't Hooft mass scale, and L ik ≡ log (M 2 i /M 2 k ) has been defined. Note that the fermion triplet has Y = 0, so no threshold correction to the gauge coupling g 1 appears at one-loop.

Matching condition for the dim-5 operator
The Wilson coefficient of the dim-5 operator at the one-loop level is given by where "Trans." denotes the transpose of the preceding term.

Matching conditions for the dim-6 operators
For different types of dim-6 operators, the matching conditions for their Wilson coefficients at the one-loop level are summarized as follows • H 4 D 2 qq .
• H 6 • ψ 2 XH • Four-quark • Four-lepton • Semileptonic With all the matching conditions for the Wilson coefficients of the operators up to dim-6, we are able to write down the complete Lagrangian of the SEFT-III at the one-loop level, i.e., where the original parameter g ∈ {m 2 , λ, Y l , Y u , Y d , g 2 } in the SM Lagrangian is substituted by its effective counterpart g eff = g + δg eff , with δg eff provided by Eqs. (12)- (17). For the dim-5 operator in the second line of Eq. (53), both the tree-level and one-loop-level coefficients are included, i.e., C eff tree + C (5) eff loop , which are respectively given by Eq. (6) and Eq. (19). In addition, three dim-6 operators from the tree-level matching are shown separately in the second line and their Wilson coefficients are given in Eq. (9). In the last line of Eq. (53), we formally sum up all the loop-induced dim-6 operators in the Warsaw basis, as explicitly shown in Table 1, and their coefficients are given in Eqs. (20)-(52).

Comparison with previous results
The one-loop matching results of the type-III seesaw model should be compared with the previous ones in Ref. [48], which are obtained in assumption of the mass degeneracy for three fermionic triplets. To make such a comparison easier, we take the equal-mass limit of our general results in Eq. (53), i.e., M 1 = M 2 = M 3 ≡ M , and find that most of our results are consistent with those in Ref. [48]. However, we do observe some mistakes in Ref. [48]. More explicitly, the matching conditions for the coefficients {δλ, δY l , C eH , C (1) Hl } in the equal-mass limit should be corrected as follows where L ≡ log (µ 2 /M 2 ) has been defined and N Σ denotes the number of fermionic triplets. Two further comments are in order. First, note that our notation for Y Σ is actually the Hermitian conjugation of that in Ref. [48], and m 2 = −µ 2 H is implied. Second, it should be pointed out that the effects of field normalization on δλ and δY l haven't been considered in the above equations in order to perform a direct comparison with Ref. [48]. Additionally, an overall factor of 1/(16π 2 ) has been dropped in these coefficients for the same reason.

Radiative decays of charged leptons
The one-loop matching conditions derived in the previous section can be implemented to carry out self-consistent one-loop calculations in the SEFT-III, which are supposed to reproduce the same results in the UV full theory, i.e., the type-III seesaw model. As an explicit example, we l − process µ − (p 1 ) → e − (p 2 ) + γ(q), we can compute the amplitudes for the diagrams in Fig. 1 M , where U = (1 + RR † /2)U 0 is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix for lepton flavor mixing with U 0 being the lowest-order unitary mixing matrix, G F / √ 2 = g 2 2 /(8M 2 W ) is the Fermi constant, andM ν ≡ diag{m 1 , m 2 , m 3 } with m i (for i = 1, 2, 3) being neutrino masses. It is easy to verify that U U † ≃ 1 + RR † and thus the PMNS matrix is non-unitary. 1 Consequently, the decay rate in the SEFT-III up to O(M −2 Σ ) is given by On the other hand, one can also compute the decay rate in the full type-III seesaw model, as has been done in Ref. [57]. Based on the calculations in Ref. [57], we expand the results therein with respect to 1/M Σ and retain the terms up to O(M −2 Σ ). The final result reads where the coefficient C = −6.56 is given in Ref. [57]. Given s 2 w ≈ 0.23, it is straightforward to verify the coefficient 16(c 2 w − 2)/3 ≈ −6.56 in Eq. (57). As expected, Eq. (57) agrees perfectly with Eq. (58). Therefore, starting with the EFT Lagrangian alone, one will be able to carry out complete one-loop calculations of the low-energy observables in a similar way as the simple example shows in this subsection. It is worthwhile to stress that the dim-6 operators O eW and O eB arising from the one-loop matching play an important role [58].

Beta function of the quartic Higgs coupling
In the SM, around the energy scale µ = O(10 11 ) GeV, the running quartic Higgs coupling λ(µ) becomes negative [59,60], rendering the electroweak vacuum to be unstable. The main reason why λ(µ) declines rapidly with the increasing µ is that the beta function of λ contains a large negative contribution from the top Yukawa coupling y t . In the type-III seesaw model, this problem will be more serious, since λ decreases faster than it does in the SM due to the existence of the Yukawa coupling Y Σ of the fermionic triplet. The beta function of λ in the type-III seesaw model is [61]: where the beta function β SM λ in the SM has been identified in the last line. Since we have matched the type-III seesaw model onto the SMEFT, it is interesting to clarify the relationship between the beta function in the UV full theory and that derived in the EFT. From the EFT perspective, the one-loop renormalization-group (RG) running of λ EFT can be studied in the traditional way [30]. More explicitly, in the SEFT-III where heavy fermionic triplets have been integrated out, the running of λ EFT can be triggered by higher-dimensional operators apart from original SM contributions [62,63]: where it is sufficient to input the tree-level results of the Wilson coefficients in Eq. (9) for selfconsistent computations at one-loop level. Note that λ EFT ̸ = λ here, because the UV theory and the EFT may differ in the UV-divergent behaviors (see, e.g., Ref. [64], for a review).
However, since we are working on a UV-motivated EFT, we can not only maintain the infrared (IR) information but also even reconstruct the beta functions of the UV theory under certain conditions. In fact, according to the EFT running in Eq. (60) and one-loop matching result, one can reproduce the running behavior of λ in the full type-III seesaw model in Eq. (59). The key point is that we need to add the contribution from the threshold correction to λ into the beta function in Eq. (60) in the following way where δλ eff is the threshold correction to λ given in Eq. (17) and it depends on the renormalization scale µ. Extracting the µ-dependence of δλ eff in Eq. (17) explicitly, one can check that the equality in Eq. (61) indeed holds. With the above demonstration, we emphasize that the complete expression of β(λ) in the full theory can be reproduced by summing up the threshold contribution and the beta function β(λ EFT ) in the EFT. The result in Eq. (61) can be easily understood by focusing on the divergent behaviors from the perspective of the region expansion in the UV theory. First, note that the beta function β(λ EFT ) in the EFT is governed by the UV divergence of the soft region of loop momentum in the UV theory. However, this UV divergence can be offset by the IR divergence of the hard region loop momentum, the UV divergence of which corresponds to the true one of the UV full theory. Because of this, the µ-dependence of δλ eff in Eq. (61) comes from two sources, i.e., the UV and IR divergences. Consequently, the IR contribution in 16π 2 µd(δλ eff )/dµ will cancel β(λ EFT ) out. In the meanwhile, the remaining part from the UV contribution just produces β(λ). In this way, Eq. (61) will hold for a general UV theory and its EFT description.

Strategy to distinguish among SEFTs
Thus far all three types of seesaw models have been matched to the SMEFT at the one-loop level. As for the effective operators up to dim-6, these three SEFTs show clear differences between each other, as indicated in Table 1. Therefore, a natural and interesting question is whether these different dim-6 operators can be implemented to experimentally distinguish among three types of seesaw models. To fully answer this question, one must perform a global-fit analysis of all existing experimental measurements in the framework of the SEFTs and make a model comparison. In this subsection, we just outline a preliminary strategy to look for the answer.
First of all, we notice that the differences among three SEFTs mainly appear as the four-fermion operators. From Table 1, there are 31 dim-6 operators in the SEFT-I, whereas two additional ones (namely, O W and O (3) qq ) exist in the SEFT-III. As for the SEFT-II, eight more dim-6 operators are present, including O (1) qq , O qe and six four-fermion operators of type RRRR, when compared to the SEFT-III. Based on these observations, we propose to search for physical observables that are sensitive to the four-fermion operators in collider experiments. The global-fit analysis of fourfermion operators in the SMEFT indicates that the data from the top-quark sector at the CERN Large Hadron Collider (LHC) may provide very useful information [65,66]. For instance, the toppair production is sensitive to four-quark operators, but there occur many degeneracies among these operators and more observables will be helpful to break the degeneracies [67]. Motivated by these studies for the SMEFT, we shall focus on the processes also in the top-quark sector.
Then, we suggest looking into the single-top production at high-energy hadron colliders. As shown in Ref. [68], the impact of the dim-6 operator O qq on the single-top production is significant. This operator appears in both the SEFT-II and the SEFT-III, but not in the SEFT-I, implying the possibility to distinguish the latter one from the former two. More explicitly, three operators O Hq at dim-6 in the SMEFT contribute to the single-top production mainly through the interference with the SM contribution. However, O uW will be neglected in our analysis because it is absent in all three SEFTs. The rest two operators contribute to the production cross-sections of u +d → t +b in the s-channel and u + b → d + t in the t-channel [68]  qq is found to be nonzero, the SEFT-I can be immediately excluded, since it does not contain this operator.
To estimate the experimental sensitivity to these two relevant Wilson coefficients, we utilize a simple statistical analysis by constructing the χ 2 -function as follows where σ s exp. (or σ t exp. ) stands for the measured cross-section in the s-(or t-) channel at the LHC [69] and likewise σ s EFT (or σ t EFT ) for the expected cross-section in the EFT. While δσ s and δσ t are the uncertainties of experimental measurements, the theoretical uncertainty of the EFT crosssection has been taken into account by introducing a nuisance parameter f and its error δf as a penalty term into the χ 2 function. To obtain the total hadronic cross-section in the protonproton colliders, one has to convolve the partonic cross-sections in Eq. (62) with the parton distribution functions (PDF) with the energy threshold being M t . For this purpose, we adopt the NNPDF31_nlo_as_0118_luxqed PDFs [70], and the s-and t-channel hadronic cross-sections under the V tb → 1 approximation are where σ s SM and σ t SM are the SM cross-sections for the s-channel and the t-channel, respectively. From the LHC run-II data at the center-of-mass energy √ s = 8 TeV with the total luminosity L = 20.2 fb −1 , the measured single-top cross-sections are σ s exp. = 4.8 +1.8 −1.5 pb and σ t exp. = 89.6 +7.1 −6.3 pb, respectively, whereas the NLO+NNLL SM predictions are 5.61 ± 0.22 pb and 87.8 +3.4 −1.9 pb [71]. Given these input values, a simplified version of the χ 2 -fit analysis can be accomplished by imposing a constraint χ 2 − χ 2 min ≤ ∆χ 2 at the 68% and 95% confidence levels. The final results of the χ 2 -fit analysis are shown in Fig. 2. At the the 95% confidence level, we obtain the allowed regions C  Fig. 2 into the lower bounds on the scales of the type-II and type-III seesaw models It can be seen that these lower bounds on the seesaw scales are rather weak, because the coefficients from one-loop matching are suppressed and the current measurements are not precise enough. In addition, Fig. 2 shows that the LHC run-II data are well consistent with the SM predictions. However, the situation may be greatly improved for future collider experiments. For example, at the HL-LHC, the luminosity will be increased to 3000 fb −1 , such that the experimental uncertainty will be reduced to one percent of the present one. With such a high precision, it is hopefully possible to detect visible deviations from the SM predictions, and the information about the typical scales of type-II and type-III seesaw models can be obtained. Finally, we point out that further discrimination between the SEFT-II and the SEFT-III is possible but more challenging. When √ C/Λ ∼ 1 TeV −1 is sizable with C being a general Wilson coefficient of the dim-6 operator, we also need to include the dim-6 squared terms, apart from the interference terms, into the global fit for the single-top production. In this case, there are  [67]. Fortunately, the last two operators are absent in the SEFT-II and SEFT-III, as shown in Table 1, so only O (1) qq is relevant. As this operator happens to be in the SEFT-II, but not in the SEFT-III, it can be utilized to further distinguish between the SEFT-II and the SEFT-III in the measurements of single-top production. Once the dim-6 squared terms are added into the analysis, any indication of nonzero values of C (1) qq will be a smoking-gun signal for the type-II seesaw model, while excluding the type-I and type-III seesaw models. However, a detailed analysis along this line is beyond the scope of this paper and will be left for future works.

Conclusions
In the present paper, we have accomplished the complete one-loop matching of the type-III seesaw model onto the SMEFT via both functional and diagrammatic approaches. The general results for three generations of heavy fermionic triplets are given. A careful comparison with the previous results in Ref. [48], where the equal-mass limit has been taken for the fermionic triplets, indicates that some mistakes in the Wilson coefficients in Ref. [48] need to be corrected. The correct results are also summarized in Sec. 3.4. Furthermore, the low-energy phenomenology of the SEFT-III in three aspects is explored. First, we calculate the rates of lepton-flavor-violating decays of charged leptons in the SEFT-III and demonstrate that the results in the full type-III seesaw model in the large-mass limit can indeed be reproduced when the one-loop matching operators and the associated Wilson coefficients are taken into account. This is a simple example for self-consistent one-loop calculations in the SEFT-III, which will be important to probe neutrino mass models in the precision era of particle physics. Then, we investigate the relationship between the beta function β(λ EFT ) of the running quartic Higgs coupling λ EFT in the SEFT-III and that β(λ) in the full type-III seesaw model. It has been shown that β(λ) in the full theory can be derived by summing up β(λ EFT ) in the EFT and the contribution from the µ-dependence of one-loop matching condition for λ. All these discussions manifest the importance of one-loop matching for phenomenological studies in the EFT. Finally, with the EFTs for three types of seesaw models, we propose a possible way to distinguish among them in collider experiments. For example, the single-top production in the hadron colliders is sensitive to dim-6 four-fermion operators. A novel way to rule out the SEFT-I is to discover the contribution from the operator O (3) qq , which is absent in the SEFT-I. The further discrimination between the SEFT-II and the SEFT-III relies on the observation of the dim-6 operator O (1) qq , which is more challenging. It is interesting to see whether this strategy can be really implemented in the future analysis of the HL-LHC data to single out the true mechanism for neutrino mass generation.
The origin of neutrino masses definitely calls for new physics beyond the SM. As the canonical seesaw models for neutrino masses usually work at the energy scale far beyond the electroweak scale, precision calculations in their low-energy EFTs will be indispensable when more accurate data are available at the high-energy and high-intensity frontiers. We believe that the one-loop construction of the SEFTs for three types of canonical seesaw models will be very useful for their phenomenological studies at low energies in a systematic way.