# Radiative Decays of Charged Leptons in the Seesaw Effective Field Theory with One-loop Matching

RRadiative Decays of Charged Leptons in the SeesawEﬀective Field Theory with One-loop Matching

Di Zhang a, b ∗ , Shun Zhou a, b † a Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China b School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Abstract

The canonical type-I seesaw model with three heavy Majorana neutrinos is one of themost natural extensions of the standard model (SM) to accommodate tiny Majorana massesof three ordinary neutrinos. At low-energy scales, Majorana neutrino masses and unitarityviolation of lepton ﬂavor mixing have been extensively discussed in the literature, which arerespectively generated by the unique dimension-ﬁve Weinberg operator and one dimension-six operator in the seesaw eﬀective ﬁeld theory (SEFT) with the tree-level matching. In thiswork, we clarify that a self-consistent calculation of radiative decays of charged leptons β − → α − + γ requires the SEFT with one-loop matching, where new six-dimensional operatorsemerge and make important contributions. For the ﬁrst time, the Wilson coeﬃcients of allthe relevant six-dimensional operators are computed by carrying out the one-loop matchingbetween the eﬀective theory and full seesaw model, and applied to calculate the total ratesof radiative decays of charged leptons. ∗ E-mail: [email protected] † E-mail: [email protected] (corresponding author) a r X i v : . [ h e p - ph ] F e b — In past few decades, a number of elegant neutrino oscillation experiments have ﬁrmlyestablished that neutrinos are massive and lepton ﬂavor mixing is signiﬁcant [1]. Obviously, theorigin of neutrino masses and lepton ﬂavor mixing calls for new physics beyond the standardmodel (SM). Among various new-physics extensions of the SM, the canonical type-I seesaw modelwith three heavy Majorana neutrinos is the simplest and most natural one to accommodate tinyneutrino masses [2–6]. Therefore, it is important and necessary to explore the phenomenologicalconsequences of the seesaw model in all aspects, and confront the theoretical predictions withexperimental observations. However, the experimental tests of the seesaw model depend verymuch on the absolute scale of heavy Majorana neutrino masses and their interaction strength withthe SM particles. In the original version of type-I seesaw model, heavy Majorana neutrino massesare supposed to be close to the energy scale of grand uniﬁed theories (GUT), namely, Λ GUT =2 × GeV. Although such heavy Majorana neutrinos are impossible to be directly observed inthe terrestrial experiments, they can play a crucial role in the early Universe, oﬀering an intriguingexplanation for cosmological matter-antimatter asymmetry via the leptogenesis mechanism [7]. Atlow-energy scales, it is in practice more convenient to construct an eﬀective theory of the seesawmodel, which will be henceforth called Seesaw Eﬀective Field Theory (SEFT), such that theimpact of heavy Majorana neutrinos on low-energy physics is incorporated into a series of ﬁeldoperators of mass dimensions higher than four. It is evident that a consistent derivation of higher-dimensional operators is indispensable for us to extract correct information on the seesaw modelfrom low-energy experimental data.As pointed out by Weinberg long time ago [8], if the SM is viewed as an eﬀective theoryat the electroweak scale and higher-dimensional operators are included, there exists one uniquedimension-ﬁve operator O (5) = (cid:96) L (cid:101) H (cid:101) H T (cid:96) cL that leads to the Majorana mass term of light neutrinos,where (cid:96) L and H are respectively the left-handed lepton doublet and the Higgs doublet, and (cid:96) cL ≡ C (cid:96) LT and (cid:101) H ≡ i σ H ∗ have been deﬁned in the usual way. In general the gauge-invariantLagrangian of the standard model eﬀective ﬁeld theory (SMEFT) [9, 10] (see, e.g., Ref. [11], for arecent review) can be written as L SMEFT = L SM + (cid:88) d L ( d ) = L SM + n d (cid:88) i =1 d − C ( d ) i O ( d ) i , (1)where d > O ( d ) i (for i = 1 , , · · · , n d with n d beingthe total number of d -dimensional operators), C ( d ) i denotes the corresponding Wilson coeﬃcient,and Λ is the cutoﬀ scale that can be identiﬁed with heavy Majorana neutrino masses in theSEFT. The complete sets of the operators up to a given mass dimension, such as dim-6 [9,10], dim-7 [12,13], dim-8 [14,15] and dim-9 [16], in the SMEFT or its extension with sterile neutrinos [17–20]have been constructed and the one-loop renormalization group equations (RGEs) for the dim-5 [21], dim-6 [22–24] and dim-7 [25] operators have been derived. On the other hand, for a speciﬁcrenormalizable model at a high-energy scale, one can match it to the (SM)EFT to study itslow-energy consequences. There are only few available examples of complete or partial one-loopmatching for the SM extended with a charged scalar singlet [26], a real scalar singlet [27–31], a realscalar triplet [32–34], a vector-like quark singlet [35], a light sterile neutrino and heavy fermionsand a scalar singlet [36], and two scalar leptoquarks [37] by either diagrammatic calculations or2unctional approaches [28,31–34,38–44]. Though some attempts [18,19,45–62] have also been madeto examine the various aspects of the seesaw phenomenology by using EFT techniques, to our bestknowledge, the SEFT with one-loop matching has never been accomplished even in a speciﬁc case.As far as the tree-level matching between the SEFT with operators of d ≤ O (6) = ( (cid:96) L (cid:101) H )i /∂ ( (cid:101) H † (cid:96) L ) should be taken into account [45, 46]. It is well known that the Weinbergoperator is responsible for neutrino masses and lepton ﬂavor mixing, whereas such a dimension-sixoperator actually induces the unitarity violation of the lepton ﬂavor mixing matrix [49].In this work, we initiate the complete one-loop matching for the SEFT up to dimension-six operators, and take the radiative decays of charged leptons β − → α − + γ , where ( α, β )runs over ( e, µ ), ( e, τ ) and ( µ, τ ), as a well-motivated example to demonstrate the diagrammaticprescription. The complete one-loop matching for SEFT will be done by both diagrammaticcalculations and functional approaches in the forthcoming separate works [63]. The experimentalsearches for radiative β − → α − + γ decays have placed the ever most stringent bound on leptonﬂavor violation in the charged-lepton sector, and the discovery of such rare decays will deﬁnitelypoint to new physics beyond the SM [64–66]. The null signal of β − → α − + γ decays has beenused to constrain the unitarity violation of Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixingmatrix [67–69] in both the minimal unitarity violation (MUV) scheme (with only one tree-leveldim-6 operator besides the unique dim-5 operator) [49, 58] and the type-I seesaw model [70, 71].The discrepancy between the constraints in these two cases has been noticed [70, 71]. In fact, onehas to perform the one-loop matching and take into account new dimension-six operators emergingonly at the one-loop level, as will be shown and clariﬁed in this work. For this purpose, we putforward a practically eﬃcient method to derive a complete set of higher-dimensional operatorsrelevant for a given physical process and determine the associated Wilson coeﬃcients. Althoughthe general principles for the construction of eﬀective theories can be found in the literature,technical diﬃculties usually show up case by case. — In order to derive the operators up to dimension-six at the one-loop level, we ﬁrst writedown the Lagrangian of the full theory, i.e., the type-I seesaw model L full = L SM + N R i /∂N R − (cid:18) N cR M N R + (cid:96) L Y ν (cid:101) HN R + h . c . (cid:19) , (2)where N i R (for i = 1 , ,

3) are three right-handed neutrino singlets. Without loss of generality,we work in the ﬂavor basis where both the charged-lepton Yukawa coupling matrix Y l and theMajorana mass matrix M = Diag { M , M , M } of right-handed neutrinos are diagonal. With thetree-level matching [45, 46], it is straightforward to obtain the Wilson coeﬃcients of the Weinbergoperator O (5) αβ = (cid:96) α L (cid:101) H (cid:101) H T (cid:96) c β L and the dimension-six operator O (6) αβ = ( (cid:96) α L (cid:101) H )i /∂ ( (cid:101) H † (cid:96) β L ), namely, C (5) αβ = Λ( Y ν M − Y T ν ) αβ / C (6) αβ = Λ ( Y ν M − Y † ν ) αβ , where the ﬂavor indices have been explicitlyshown. Notice that this dimension-six operator appearing at the tree level is related to twodimension-six operators in the Warsaw basis [10] by( (cid:96) α L (cid:101) H )i /∂ ( (cid:101) H † (cid:96) β L ) = 14 (cid:104)(cid:0) (cid:96) α L γ µ (cid:96) β L (cid:1) (cid:16) H † i ↔ D µ H (cid:17) − (cid:0) (cid:96) α L γ µ τ I (cid:96) β L (cid:1) (cid:16) H † i ↔ D Iµ H (cid:17)(cid:105) , (3)in which ↔ D µ ≡ D µ − ← D µ and ↔ D Iµ ≡ τ I D µ − ← D µ τ I with τ I (for I = 1 , ,

3) being the Pauli matrices3nd ← D µ acting on the left. The covariant derivative D µ is deﬁned as D µ ≡ ∂ µ − i g Y B µ − i g T I W Iµ where Y and T I (for I = 1 , ,

3) are the generators of the SM gauge groups U(1) Y and SU(2) L ,respectively, while g and g are the corresponding gauge coupling constants. This dimension-sixoperator will modify the couplings of neutrinos with both W and Z gauge bosons, resulting in theunitarity violation of the PMNS matrix.We focus on the radiative decays of charged leptons β − → α − + γ , which are lepton-ﬂavor-violating and deﬁnitely occur in nature in light of nonzero neutrino masses. On the one hand, thetree-level SEFT Lagrangian (i.e., the SM Lagrangian together with the operators O (5) αβ and O (6) αβ arising from the tree-level matching) contributes to the radiative decays of charged leptons viaone-loop diagrams mediated by massive neutrinos. On the other hand, among all the dimension-six operators in the Warsaw basis [10], only two operators O eB,αβ = (cid:0) (cid:96) α L σ µν E β R (cid:1) HB µν and O eW,αβ = (cid:0) (cid:96) α L σ µν E β R (cid:1) τ I HW Iµν make direct contributions at one-loop level. Therefore, we haveto calculate the Wilson coeﬃcients of those two operators via one-loop matching. The basicstrategy to achieve this goal is outline as below. • First, one should compute relevant one-light-particle-irreducible diagrams at the one-looplevel, and perform the matching between the SEFT and the full theory, where all the externalparticles are taken to be oﬀ-shell. In this matching process, one can obtain the operators in aredundant basis, the so-called Green’s basis [29] (see also Ref. [37]), after applying algebraic,Fierz identities and integration by parts. • Second, further making use of equations of motion (EOMs), one may convert the redundantoperators in the Green’s basis to the independent operators in the Warsaw basis, and thusget the corresponding Wilson coeﬃcients for the operators in the Warsaw basis.In practice, to calculate the Wilson coeﬃcients of O eB,αβ and O eW,αβ relevant to the radiativedecays of charged leptons, one needs to ﬁnd out the operators and the associated Wilson coeﬃcientsin the Green’s basis, which can be converted into O eB,αβ and O eW,αβ by applying algebraic, Fierzidentities, integration by parts and EOMs. Based on the complete list of dimension-six operators inRef. [10], we ﬁnd that those in classes H D , ψ D , X D , ψ HD , ψ XD and ψ XH (here ψ , X and D stand for fermion ﬁelds, gauge ﬁeld strength tensors and covariant derivative, respectively)are relevant. Then, with the help of the package BasisGen [72], we generate a series of dimension-six operators in these classes, where the redundancies due to algebraic or Fierz transformationsor integration by parts have been removed but EOMs have not been applied. In this basis, theoperators of our interest are listed in Table 1. Some helpful comments are in order. • Since right-handed neutrinos N R only interact with the SM lepton and Higgs doublets via theYukawa coupling term, the operators O eD , O JD (for J = B, W ) and O [ i ] eBD (for i = 1 , , E R or gauge bosons (i.e., B and W ) cannotbe generated by right-handed neutrinos at the one-loop level and the corresponding Wilsoncoeﬃcients are simply vanishing. • Even if the EOMs of H , B and W are taken into account, the operators O H , O [1] eH and O [1] (cid:96)JD (for J = B, W ) cannot be reduced to O eJ (for J = B, W ), implying that they are irrelevant4lasses Operators H D O H = ( D H ) † ( D H ) ψ D O (cid:96)D,αβ = i2 (cid:96) α L (cid:0) D /D + /DD (cid:1) (cid:96) β L , O eD,αβ = i2 E α R (cid:0) D /D + /DD (cid:1) E β R X D O BD = D ρ B µν D ρ B µν , O W D = ( D ρ W µν ) I (cid:0) D ρ W µν (cid:1) I ψ HD O [1] eH,αβ = (cid:96) α L E β R D H , O [2] eH,αβ = (cid:96) α L D µ E β R D µ H O [3] eH,αβ = (cid:96) α L D E β R H , O [4] eH,αβ = i (cid:96) α L σ µν D µ E β R D ν H ψ XD O [1] (cid:96)BD,αβ = (cid:96) α L γ µ (cid:96) β L D ν B µν , O [2] (cid:96)BD,αβ = i2 (cid:96) α L γ µ ↔ D ν (cid:96) β L B µν O [3] (cid:96)BD,αβ = i2 (cid:96) α L γ µ ↔ D ν (cid:96) β L (cid:101) B µν , O [1] eBD,αβ = E α R γ µ E β R D ν B µν O [2] eBD,αβ = i2 E α R γ µ ↔ D ν E β R B µν , O [3] eBD,αβ = i2 E α R γ µ ↔ D ν E β R (cid:101) B µν O [1] (cid:96)W D,αβ = (cid:96) α L γ µ τ I (cid:96) β L (cid:0) D ν W µν (cid:1) I , O [2] (cid:96)W D,αβ = i2 (cid:96) α L γ µ ↔ D Iν (cid:96) β L W Iµν O [3] (cid:96)W D,αβ = i2 (cid:96) α L γ µ ↔ D Iν (cid:96) β L (cid:102) W Iµν ψ XH O eB,αβ = (cid:0) (cid:96) α L σ µν E β R (cid:1) HB µν , O eW,αβ = (cid:0) (cid:96) α L σ µν E β R (cid:1) τ I HW Iµν

Table 1: The dimension-six operators in classes H D , ψ D , X D , ψ HD , ψ XD and ψ XH ,where the Hermitian conjugates of the operators in classes ψ HD and ψ XH are not listedexplicitly and the most relevant operators for one-loop matching are highlighted in boldface. Inaddition, α and β (for α, β = e, µ, τ ) are lepton ﬂavor indices, D ρ B µν ≡ ∂ ρ B µν , (cid:0) D ρ W µν (cid:1) I ≡ ∂ ρ W Iµν + g ε IJK W Jρ W Kµν , while (cid:101) X µν (for X = B, W I ) are the dual gauge ﬁeld strength tensorsdeﬁned as (cid:101) X µν = ε µνρσ X ρσ / ε = +1.to radiative decays of charged leptons. However, O [1] eH has to be considered when calculatingthe Wilson coeﬃcients of other relevant operators, especially those of O eJ (for J = B, W ).In summary, only the operators O (cid:96)D , O [ i ] eH , O [ j ] (cid:96)JD and O eJ (for i = 1 , , , j = 2 , J = B, W ), which have been highlighted in boldface in Table 1, need to be considered. Oncewe obtain their Wilson coeﬃcients, those of O eJ (for J = B, W ) in the Warsaw basis can bederived by using algebraic, Fierz identity, integration by parts and the EOMs. This leads us tothe complete set of relevant one-loop-level operators with correct Wilson coeﬃcients. Togetherwith the tree-level SEFT Lagrangian, these operators generate the full amplitude of radiativedecays of charged leptons, with which the decay rates can be calculated. — Now we present explicit one-loop matching between the SEFT working in the infraredregion (IR) and the full theory in the ultraviolet region (UV), and ﬁgure out the Wilson coeﬃcientsof relevant operators at one-loop level by using the approach of Feynman diagrams. All thecalculations will be performed with the dimensional regularization and the modiﬁed minimalsubtraction (MS) scheme for the space-time dimension of d ≡ − (cid:15) . The basic idea is toutilize the method of expansion by regions [73–75]. More explicitly, for a given one-loop Feynman5 ... . B µ (or W Iµ ) qE β p ℓ aα p H b p H e H d ℓ cγ N i k (a) . ... . E β p ℓ aα p H b p B µ (or W Iµ ) q ℓ cγ ℓ dγ N i H e k (b) Figure 1: Feynman diagrams for the amplitude (cid:104) (cid:96)EHJ (cid:105) in the ultraviolet (UV) full theory, whichshould be matched by the operators O [ i ] eH (for i = 1 , , ,

4) and O eJ (for J = B, W ) in the infrared(IR) eﬀective theory.amplitude in the UV theory, one can split it into hard- (i.e., k ∼ M i (cid:29) p with k , M i and p standing for the loop momentum, heavy particle masses, and external momenta, respectively) andsoft-momentum (i.e., k ∼ p (cid:28) M i ) regions in dimensional regularization. In each region, theintegrand of the loop integral is expanded as a Taylor series with respect to the parameters thatare considered small, and then the integrand should be integrated over the whole d -dimensionalspace of the loop momentum (i.e., k ). Then, the Wilson coeﬃcients of the operators at one-looplevel can be identiﬁed by equating the contributions from the hard-momentum region with thosefrom the one-loop operators. The contributions from the soft-momentum region are exactly thesame as those from the tree-level SEFT Lagrangian via one-loop diagrams. Consequently, as forone-loop matching, we pay attention only to the hard-momentum region of the amplitudes in theUV and the contributions from the one-loop operators in the IR.The operators O [ i ] eH (for i = 1 , , ,

4) and O eJ (for J = B, W ) can be matched by computingthe amplitudes given by the Feyman diagrams (a) and (b) in Fig. 1. We ﬁrst consider the one-loopmatching for O [ i ] eH (for i = 1 , , ,

4) and O eB . The corresponding amplitudes for diagrams (a) and(b) for B µ in the UV arei M B, UV a = u ( p ) P R (cid:90) d k (2 π ) (cid:16) /k + /q + /p (cid:17) (cid:16) /k + /p (cid:17) (2 k + q ) µ k ( k + q ) ( k + p ) (cid:2) ( k + q + p ) − M i (cid:3) u ( p ) (cid:15) ∗ µ ( q ) × g δ ab ( Y ν ) αi (cid:0) Y † ν Y l (cid:1) iβ , (4)and i M B, UV b = u ( p ) P R (cid:90) d k (2 π ) (cid:16) /k + /p (cid:17) (cid:16) /k + /p − /q (cid:17) γ µ (cid:16) /k + /p (cid:17) k ( k + p ) ( k + p − q ) (cid:2) ( k + p ) − M i (cid:3) u ( p ) (cid:15) ∗ µ ( q ) × (cid:18) − (cid:19) g δ ab ( Y ν ) αi (cid:0) Y † ν Y l (cid:1) iβ , (5)6here the repeated indices are summed. The hard-momentum parts of these two amplitudes canbe obtained by expanding the integrands in the limit of p (cid:28) k, M i with p being any externalmomentum (i.e., p = p , p , p , q in this case). Since the combination (cid:96)EHB of relevant ﬁeldsis already of mass-dimension ﬁve, the relevant terms should be proportional to the ﬁrst power ofthe external momentum, which is equivalent to one space-time derivative of mass-dimension one.With the help of Eqs. (4) and (5), we can get the contributions from the hard-momentum regioni M B, UV a | hard = i g δ ab ( Y ν ) αi (cid:0) Y † ν Y l (cid:1) iβ π ) M i u ( p ) P R (cid:26) (cid:104) γ µ (cid:16) /p − /p − /q (cid:17) − p − p − q ) µ (cid:105) ln µ M i + 3 γ µ (cid:16) /p − /p − /q (cid:17) − p − p − q ) µ (cid:111) u ( p ) (cid:15) ∗ µ ( q ) , (6)andi M B, UV b | hard = − i g δ ab ( Y ν ) αi (cid:0) Y † ν Y l (cid:1) iβ π ) M i u ( p ) P R (cid:26) (cid:104) γ µ (cid:16) /p + /q (cid:17) + 2 ( p − p − q ) µ (cid:105) ln µ M i + γ µ (cid:16) /p + /q (cid:17) + 2 ( p − p − q ) µ (cid:111) u ( p ) (cid:15) ∗ µ ( q ) , (7)where only the terms proportional to p = p , p , p , q have been retained and µ is the renormal-ization scale. The divergences have not been explicitly shown but they can be easily recoveredby setting ln( µ /M i ) → ln( µ /M i ) + ∆ (cid:15) , where ∆ (cid:15) ≡ /(cid:15) − γ E + ln (4 π ) with γ E being the Eulerconstant. Then the hard-momentum part of the total amplitude readsi M B, UVtot | hard = i M B, UV a | hard + i M B, UV b | hard = i g δ ab ( Y ν ) αi (cid:0) Y † ν Y l (cid:1) iβ π ) M i u ( p ) P R (cid:26) (cid:104) γ µ (cid:16) /p − /p − /q (cid:17) − p − p − q ) µ (cid:105) ln µ M i + γ µ (cid:16) /p − /p − /q (cid:17) − p − p − q ) µ (cid:111) u ( p ) (cid:15) ∗ µ ( q ) . (8)On the other hand, the contribution of one-loop operators O [ i ] eH (for i = 1 , , ,

4) and O eB in theIR to the amplitude (cid:104) (cid:96)EHB (cid:105) is given byi M B, EFTtot | loop = i2 g δ ab u ( p ) P R (cid:34) γ µ /q (cid:18) g C eB − C [4] eH (cid:19) αβ + γ µ (cid:16) /p − /p (cid:17) (cid:16) C [4] eH (cid:17) αβ + q µ (cid:18) − g C eB − C [1] eH + 2 C [2] eH − C [3] eH + 2 C [4] eH (cid:19) αβ + p µ (cid:16) C [1] eH − C [2] eH + 4 C [3] eH − C [4] eH (cid:17) αβ + 2 p µ (cid:16) − C [1] eH + C [2] eH + C [4] eH (cid:17) αβ (cid:21) , (9)where C [ i ] eH (for i = 1 , , ,

4) and C eB stand for the Wilson coeﬃcients of O [ i ] eH (for i = 1 , , ,

4) and O eB , respectively. Matching at the energy scale µ = O ( M i ) and dropping the terms proportionalto ∆ (cid:15) in Eq. (8), one arrives at an array of ﬁve linear equations for ﬁve unknown Wilson coeﬃcients7y equating Eq. (8) with Eq. (9), namely, − π ) · S = 1Λ · (cid:18) g C eB − C [4] eH (cid:19) , π ) · S = 1Λ · C [4] eH , π ) · S = 1Λ · (cid:18) − g C eB − C [1] eH + 2 C [2] eH − C [3] eH + 2 C [4] eH (cid:19) , − π ) · S = 1Λ · (cid:16) C [1] eH − C [2] eH + 4 C [3] eH − C [4] eH (cid:17) , π ) · S = 1Λ · (cid:16) − C [1] eH + 2 C [2] eH + 2 C [4] eH (cid:17) , (10)with S ≡ Y ν M − Y † ν Y l . Solving these linear equations in Eq. (10), we have C [1] eH = S Λ (4 π ) , C [2] eH = 7 S Λ π ) , C [3] eH = S Λ π ) , C [4] eH = 3 S Λ π ) , C eB = g S Λ π ) . (11)To obtain the hard-momentum part of the total amplitude (cid:104) (cid:96)EHW (cid:105) in Fig. 1, one can simplyreplace g δ ed with g τ Ide for diagram (a), and g δ cd with − g τ Icd for diagram (b) in Fig. 1, due tothe diﬀerent Feynman rules for the vertices involving B and W . Such replacements givei M W, UVtot | hard = − i g τ Iab π ) M i ( Y ν ) αi (cid:0) Y † ν Y l (cid:1) iβ u ( p ) P R (cid:26) γ µ /p ln µ M i + γ µ (cid:16) /p − /q (cid:17) − p − p − q ) µ (cid:27) u ( p ) (cid:15) ∗ µ ( q ) . (12)Meanwhile, the contribution of one-loop operators O [ i ] eH (for i = 1 , , ,

4) and O eW to the totalamplitude (cid:104) (cid:96)EHW (cid:105) in the IR is given byi M W, EFTtot | loop = i2 g τ Iab u ( p ) P R (cid:34) γ µ /q g ( C eW ) αβ − γ µ /p (cid:16) C [4] eH (cid:17) αβ − q µ (cid:18) g C eW + C [1] eH (cid:19) αβ + p µ (cid:16) C [1] eH − C [2] eH + C [4] eH (cid:17) αβ − p µ (cid:16) C [1] eH (cid:17) αβ (cid:21) . (13)Similarly, by equating Eq. (12) with Eq. (13), one can obtain π ) · S = 1Λ · g C eW , π ) · S = 1Λ · C [4] eH , π ) · S = 1Λ · (cid:18) g C eW + C [1] eH (cid:19) , π ) · S = 1Λ · (cid:16) C [1] eH − C [2] eH + C [4] eH (cid:17) , π ) · S = 1Λ · C [1] eH , (14)where one can observe that the solutions to C [ i ] eH (for i = 1 , ,

4) are identical to those given inEq. (11) as they should be. In addition, the new Wilson coeﬃcient C eW is found to be C eW = g S Λ π ) . (15)8 bβ p ‘ aα p H c k H d H e N i B µ W Iν q q (a) ‘ bβ p ‘ aα p H c k H d H e N i W Iν B µ q q (b) ‘ bβ p ‘ aα p H c k H d N i B µ W Iν q q (c) Figure 2: Feynman diagrams for the amplitude (cid:104) (cid:96)(cid:96)BW (cid:105) in the UV full theory, which should bematched by the operators O (cid:96)D and O [2] (cid:96)XD (for X = B, W ) in the IR eﬀective theory.Next we proceed with the one-loop matching of the operators O (cid:96)D and O [ i ] (cid:96)JD (for i = 2 , J = B, W ) by computing the amplitude (cid:104) (cid:96)(cid:96)BW (cid:105) corresponding to three Feynman diagrams inFig. 2. As the techniques for the computations are quite similar to those in the previous case, wejust summarize the ﬁnal result for the hard-momentum part of the total amplitude, i.e.,i M UVtot | hard = − i g g τ Iab

12 (4 π ) M i ( Y ν ) αi (cid:0) Y † ν (cid:1) iβ u ( p ) P R (cid:104) g µν (cid:16) /p − /q − /q (cid:17) + (2 p − q − q ) µ γ ν + (2 p − q − q ) ν γ µ ] u ( p ) (cid:15) ∗ µ ( q ) (cid:15) I ∗ ν ( q ) , (16)where the UV divergences completely cancel out. In the IR eﬀective theory, the contributions ofthe operators O (cid:96)D and O [ i ] (cid:96)JD (for i = 2 , J = B, W ) arei M EFTtot | loop = − i4 g g τ Iab u ( p ) P R (cid:110) ( C (cid:96)D ) αβ (cid:104) g µν (cid:16) /q + /q − /p (cid:17) + γ µ ( q + q − p ) ν + γ ν ( q + q − p ) µ (cid:105) − g (cid:16) C [2] (cid:96)BD (cid:17) αβ (cid:16) g µν /q − γ µ q ν (cid:17) + 2 g (cid:16) C [3] (cid:96)BD (cid:17) αβ × (cid:104) g µν /q − γ µ q ν + γ ν (cid:16) q µ − γ µ /q (cid:17)(cid:105) + 2i g (cid:16) C [2] (cid:96)W D (cid:17) αβ (cid:16) g µν /q − γ ν q µ (cid:17) − g (cid:16) C [3] (cid:96)W D (cid:17) αβ (cid:104) g µν /q − γ ν q µ + γ µ (cid:16) q ν − γ ν /q (cid:17)(cid:105)(cid:27) u ( p ) (cid:15) ∗ µ ( q ) (cid:15) I ∗ ν ( q ) , (17)which should be identiﬁed with Eq. (16), leading to the relevant Wilson coeﬃcients C (cid:96)D = − Λ π ) Y ν M − Y † ν , C [2] (cid:96)BD = C [3] (cid:96)BD = C [2] (cid:96)W D = C [3] (cid:96)W D = 0 . (18)It is worth pointing out that in the IR eﬀective theory, in addition to the contributions from theloop-level operators, the tree-level dimension-six operator O (6) αβ = ( (cid:96) α L (cid:101) H )i /∂ ( (cid:101) H † (cid:96) β L ) also contributesto the amplitudes (cid:104) (cid:96)EHJ (cid:105) (for J = B, W ) and (cid:104) (cid:96)(cid:96)BW (cid:105) via one-loop diagrams with an

LLHH vertex. However, this contribution is identical with the soft-momentum part of the correspondingamplitudes in the UV full theory, which is calculated by expanding the heavy Majorana neutrinopropagator i / ( /k − M i ) in terms of k/M i . This expansion converts the nonlocal propagator intolocal terms and leads to an equivalent LLHH vertex. Thus, these two contributions are irrelevantto the one-loop matching as we have mentioned at the beginning of this section.Finally, we make use of the EOMs to transform the resultant dimension-six operators intothose in the Warsaw basis and identify the corresponding Wilson coeﬃcients of the latter. For9adiative decays of charged lepton, only O eJ (for J = B, W ) that are contained in the Warsawbasis make direct contributions. Hence we concentrate on the operators O (cid:96)D , O [ i ] eH and O [ j ] (cid:96)JD ,which are related to O eJ (for i = 2 , , j = 2 , J = B, W ) via EOMs and important for thedetermination of the Wilson coeﬃcients, while it is unnecessary to consider O [ j ] (cid:96)JD (for j = 2 , J = B, W ) for their Wilson coeﬃcients are vanishing as shown in Eq. (18). After some algebraiccomputations, we ﬁnd O [2] eH,αβ ⇒ (cid:96) α L i σ µν E β R (cid:2) D µ , D ν (cid:3) H = 18 (cid:0) g O eB,αβ + g O eW,αβ (cid:1) , (19) O [3] eH,αβ ⇒ (cid:96) α L i σ µν (cid:2) D µ , D ν (cid:3) E β R H = − g O eB,αβ , (20) O [4] eH,αβ ⇒ −O [2] eH,αβ ⇒ − (cid:0) g O eB,αβ + g O eW,αβ (cid:1) , (21) O (cid:96)D,αβ ⇒ − (cid:96) α L (cid:104) σ µν (cid:0) g B µν − g τ I W Iµν (cid:1) i /D − i ← /Dσ µν (cid:0) g B µν − g τ I W Iµν (cid:1)(cid:105) (cid:96) β L = −

18 ( Y l ) βγ (cid:0) g O eB,αγ − g O eW,αγ (cid:1) − (cid:16) Y † l (cid:17) γα (cid:0) g O eB,βγ − g O eW,βγ (cid:1) † , (22)where the terms irrelevant to the coeﬃcients of O eJ (for J = B, W ) have been neglected and theEOM of (cid:96) L has been applied in the last step of Eq. (22). According to Eqs. (19)-(22), together withthe coeﬃcients derived in Eqs. (11), (15) and (18), one can obtain the ﬁnal Wilson coeﬃcients ofthe operators O eJ (for J = B, W ) in the Warsaw basis C (cid:48) eB = C eB + g C [2] eH − g C [3] eH − g C [4] eH − g C (cid:96)D Y l = g S Λ

24 (4 π ) ,C (cid:48) eW = C eW + g C [2] eH − g C [4] eH + g C (cid:96)D Y l = 5 g S Λ

24 (4 π ) , (23)with S ≡ Y ν M − Y † ν Y l . Therefore, the one-loop SEFT Lagrangian with dimension-six operatorsrelevant for radiative decays of charged leptons is given by L (6)loop = (cid:0) Y ν M − Y † ν Y l (cid:1) αβ

24 (4 π ) (cid:2) g (cid:0) (cid:96) α L σ µν E β R (cid:1) HB µν + 5 g (cid:0) (cid:96) α L σ µν E β R (cid:1) τ I HW Iµν (cid:3) + h . c . , (24)which will be used to calculate the decay rate for β − → α − + γ in the next section. It is worthstressing that these loop-level dimension-six operators should be added to the tree-level SEFTLagrangian with the Weinberg operator and the tree-level dimension-six operator. — The one-loop SEFT Lagrangian with operators up to dimension-six has been speciﬁed,and its explicit form after the spontaneous gauge symmetry breaking reads L SEFT = ν α L (cid:16) + M D M − M † D (cid:17) αβ i /∂ν β L − (cid:20) l α L ( M l ) αβ l β R + 12 ν α L ( M ν ) αβ ν c β L + h . c . (cid:21) + (cid:18) g √ l α L γ µ ν α L W − µ + h . c . (cid:19) + g θ w ν α L γ µ ν α L Z µ + g (cid:16) M D M − M † D M l (cid:17) αβ

48 (4 π ) M W ( g cos θ w − g sin θ w ) l α L σ µν l β R F µν + h . c . , (25)10 − p α − p β − ν i γq W − (a) β − p α − p ν i γqW − W − (b) β − p α − p ν i α − γqW − (c) β − p α − p γq (d) Figure 3: Feynman diagrams for radiative decays β − → α − + γ decays at one-loop level in the unitary gauge. While (a)-(c) are mediated by massive neutrinos with a non-unitary ﬂavor mixingmatrix, (d) is generated by the dimension-six operators at one-loop level.where only the relevant terms for lepton masses, ﬂavor mixing and radiative decays of chargedleptons are kept. Some explanations for our notations are helpful. First, we have the charged-lepton mass matrix M l ≡ vY l / √ v ≈

246 GeV being the vacuum expectation value of theHiggs ﬁeld, the Dirac neutrino mass matrix M D ≡ vY ν / √

2, and the eﬀective Majorana neutrinomass matrix M ν ≡ − M D M − M TD , where the neutrino mass matrix is generated by the tree-levelWeinberg operator. Second, as in the SM, M W = g v/ W -boson mass, θ w = arctan ( g /g )is the weak mixing angle, and F µν = ∂ µ A ν − ∂ ν A µ is the gauge ﬁeld strength tensor with A µ beingthe photon ﬁeld.The terms in the ﬁrst two lines of Eq. (25) appear at tree level while those in the third line atone-loop level. As before, we shall work in the ﬂavor basis where both M l = Diag { m e , m µ , m τ } and M = Diag { M , M , M } are diagonal. One can ﬁrst make a transformation ν L → V ν L with V = − RR † / R ≡ M D M − to normalize the kinetic term of left-handed neutrino ﬁelds,and then another one ν L → U ν L such that U † V M ν V T U ∗ = (cid:99) M ν = Diag { m , m , m } . After thesetransformations, the eﬀective Lagrangian becomes L SEFT = ν L i /∂ν L − (cid:18) l L M l l R + 12 ν L (cid:99) M ν ν cL + h . c . (cid:19) + (cid:18) g √ l L γ µ U ν L W − µ + h . c . (cid:19) + g θ w ν L γ µ U † U ν L Z µ − eg

12 (4 π ) M W l L σ µν RR † M l l R F µν + h . c . , (26)where the identities e = g cos θ w = g sin θ w are implemented in the third line, and the PMNSmatrix U = V U is non-unitary as U is a unitary matrix but V not. It is worthwhile to pointout that the nonunitarity of U and the ﬂavor-changing neutral-current interaction are attributedto the existence of the dimension-six operator at the tree level in Eq. (3).Thanks to nonzero neutrino masses and the nontrivial ﬂavor mixing, the leptonic charged-current interaction induces radiative decays of charged leptons via the diagrams (a)-(c) shown inFig. 3. The sum over all three amplitudes can be found in Refs. [71, 76–78], i.e.,i M abc = − i eg π ) M W (cid:88) i =1 U αi U ∗ βi (cid:18) −

56 + m i M W (cid:19) (cid:2) (cid:15) ∗ µ u ( p ) i σ µν q ν (cid:0) m α P L + m β P R (cid:1) u ( p ) (cid:3) , (27)where the repeated indices ( α, β ) = ( e, µ ), ( e, τ ), ( µ, τ ) characterize the ﬁnal and initial chargedleptons, and should not be summed over. The dimension-six operators at the one-loop level (i.e.,11hose in the third line of Eq. (26)) contribute to radiative β − → α − + γ decays via diagram (d)in Fig. 3. From the third line of Eq. (26), it is straightforward to read oﬀ the correspondingamplitude of diagram (d)i M d = i eg π ) M W (cid:0) RR † (cid:1) αβ (cid:2) (cid:15) ∗ µ u ( p ) i σ µν q ν (cid:0) m α P L + m β P R (cid:1) u ( p ) (cid:3) . (28)Therefore, the total amplitude for the radiative decays of charged leptons in the one-loop SEFTcan be obtained by adding Eq. (27) and Eq. (28), i.e.,i M tot = i M abc + i M d = − i eg π ) M W (cid:34) (cid:88) i =1 U αi U ∗ βi (cid:18) −

56 + m i M W (cid:19) − (cid:0) RR † (cid:1) αβ (cid:35) × (cid:2) (cid:15) ∗ µ u ( p ) i σ µν q ν (cid:0) m α P L + m β P R (cid:1) u ( p ) (cid:3) . (29)One can easily verify that the 3 × U R ) is approximately the upper 3 × × U that is used to diagonalize the overall 6 × U U † + RR † = V V † + RR † = + O ( M − )holds. As expected, our result in Eq. (29) is in perfect agreement with that obtained in the fulltheory [71, 76–78] in the limit of m i (cid:28) M W (cid:28) M j (for i, j = 1 , , (cid:0) β − → α − + γ (cid:1) (cid:39) α em G m β π (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i =1 U αi U ∗ βi (cid:18) −

56 + m i M W (cid:19) − (cid:0) RR † (cid:1) αβ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (30)where α em ≡ e / (4 π ) is the electromagnetic ﬁne-structure constant, G F is the Fermi constant andthe tiny ratio m α /m β (cid:28) β − → α − + ν α + ν β is given byΓ ( β − → α − + γ )Γ (cid:0) β − → α − + ν α + ν β (cid:1) (cid:39) α em π (cid:12)(cid:12)(cid:12)(cid:12) − (cid:0) U U † (cid:1) αβ − (cid:0) RR † (cid:1) αβ (cid:12)(cid:12)(cid:12)(cid:12) = 3 α em π (cid:12)(cid:12)(cid:12)(cid:0) RR † (cid:1) αβ (cid:12)(cid:12)(cid:12) , (31)where higher-order terms have been omitted and the relation U U † + RR † (cid:39) has been usedin the last step. At this point, we clarify that the MUV scheme proposed in Ref. [49] can beregarded a practically useful framework to constrain leptonic unitarity violation by experimentaldata from electroweak precision measurements, lepton-ﬂavor-violating decays of charged leptonsand neutrino oscillations without specifying the UV full theory. However, to test any UV full theoryby precision data at low-energy scales, one has to construct the eﬀective theory and determine theWilson coeﬃcients of higher-dimensional operators by performing tree- or loop-level matching ina consistent way. For radiative β − → α − + γ decays, the results in the MUV scheme correspond tothose in Eq. (29) and Eq. (31) without the − RR † / — The solid experimental evidence for neutrino masses and lepton ﬂavor mixing indicatesthat the standard model is actually incomplete and can only serve as a low-energy eﬀective theory.Unfortunately, there has been so far no direct and signiﬁcant hint for new physics in all terrestrialexperiments other than neutrino oscillations. Almost for all kinds of particle physics experiments,the primary goal is to precisely measure fundamental parameters in nature and search for possibledeviations from the SM predictions, which signify the existence of new physics. In this regard,eﬀective theories have proved to be a very powerful and convenient tool. Therefore, the theoreticalfoundations for constructing eﬀective theories and their applications to concrete problems havereceived tremendous attention recently. This is also the case for neutrino physics.In this work, we take one of lepton-ﬂavor-violating processes, i.e., radiative decays of chargedleptons, in the type-I seesaw model as an example to illustrate how to derive the low-energyseesaw eﬀective ﬁeld theory with one-loop matching. It has been explicitly shown that the one-loop processes β − → α − + γ should be studied by using the seesaw eﬀective theory with one-loopmatching. All relevant dimension-six operators and the associated Wilson coeﬃcients are derived.However, it should be noticed that the matching has been carried out at the high-energy scaleand the renormalization-group running of the Wilson coeﬃcients has been ignored. In principle,one should follow the standard procedure to decouple heavy particles of similar masses at oneenergy scale and perform the matching at this decoupling scale. Once the eﬀective theory belowthe decoupling scale is constructed, the renormalization-group equations will be implemented toevolve all physical parameters to the next scale of heavy particles. Such a procedure should berepeated until the scale of relevant experiments is reached. We leave such a complete constructionof the seesaw eﬀective theory for future works. This work was supported in part by the National Natural Science Foundation of China undergrant No. 11775231, No. 11775232, No. 11835013 and No. 12075254, and by the CAS Center forExcellence in Particle Physics.

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