Anomalous dimensions from gauge couplings in SMEFT with right-handed neutrinos

Standard Model Neutrino Effective Field Theory (SMNEFT) is an effective theory with Standard Model (SM) gauge-invariant operators constructed only from SM and right-handed neutrino fields. For the full set of dimension-six SMNEFT operators, we present the gauge coupling terms of the one-loop anomalous dimension matrix for renormalization group evolution (RGE) of the Wilson coefficients between a new physics scale and the electroweak scale. We find that the SMNEFT operators can be divided into five subsets which are closed under RGE. Our results apply for both Dirac and Majorana neutrinos. We also discuss the operator mixing pattern numerically and comment on some interesting phenomenological implications.


Introduction
The Standard Model (SM) of particle physics is an effective theory valid to some mass scale Λ. New physics at the scale Λ may address important issues like the origin of the electroweak scale, µ EW . In the SM, electroweak symmetry breaking arises from a complex fundamental Higgs scalar. Between µ EW and Λ, an effective field theory (EFT) framework can be used to describe new physics in a model independent way. In this approach, the leading terms are given by the SM, and corrections from an underlying theory beyond the SM are described by higher dimension operators, (1.1) The operators O i are SU (3) C × SU (2) L × U (1) Y invariant and are constructed only from SM fields. The Wilson coefficients (WCs) C i , that determine the size of the contribution of operators O i , can be calculated by matching the effective theory with the underlying theory.
Analyses of higher dimension operators [1] have begun anew in the study of the SM as an effective field theory. Due to the phenomenological success of the SM gauge theory and the Higgs mechanism, the most studied EFT is the Standard Model Effective Field Theory (SMEFT) [2][3][4], which respects the SM gauge symmetry with only SM field content. The one-loop renormalization group evolution (RGE) of all dimension-six operators in SMEFT have been calculated in Refs. [5][6][7].
In the SMEFT framework, new physics is considered to be heavy with Λ µ EW . However, many experiments point to new physics with a mass scale well below the electroweak scale, and many experiments to search for new light states are planned. Since these states do not appear in SMEFT, its Lagrangian must be supplemented by interactions between these new states and the SM fields. Possible new states are right-handed neutrinos that are sterile under SM gauge interactions. The masses of the sterile neutrinos can vary over a large range and can be heavy or light compared to the electroweak scale. Light sterile neutrinos have been invoked to explain many phenomena; see Ref. [8] for a review.
In this paper, we consider the sterile neutrinos to be light so that they appear as explicit degree of freedoms in the EFT framework. We use the Standard Model Neutrino Effective Field Theory (SMNEFT) which augments SMEFT with right-handed (RH) neutrinos n [9,10]. The RGE of some SMNEFT operators have been calculated. The mixing between the bosonic operators has been calculated in Ref. [11], and the one-loop RGE of a subset of four-fermion operators are given in Ref. [12]. In this work, we present the gauge terms of the one-loop RGE of all dimension-six operators in SMNEFT.
The paper is organized as follows. In section 2, we define SMNEFT and establish our notation. Our diagrammatic approach to calculate the one-loop anomalous dimension matrix (ADM) is described in section 3. In section 4, we present the ADM. In section 5, we study operator mixing using the leading-log approximation. We discuss some phenomenological implications in section 6, and summarize in section 7. Details of our calculations are provided in an appendix.

SMNEFT
In this section, we present SMNEFT. Neutrinos may be Dirac or Majorana. In the case of Dirac neutrinos, ν R ≡ n, with n and the left-handed neutrino ν L in the same spinor ν D = (ν L , n) T , and of the same mass. In the Majorana case, n and ν L are components of two different spinors, ν M = (ν L , ν c L ) T , n M = (n c , n) T , and can have different masses. Our results are valid for both cases because we focus on the gauge sector. Without specifying any possible Majorana and Dirac mass terms, the dimension-six B and L conserving SMNEFT Lagrangian is where C i are the WCs with the scale of new physics absorbed in them, and the SM Lagrangian is given by Here,φ j = jk (φ k ) * , and the Higgs vacuum expectation value is φ = v/ √ 2 with v = 246 GeV. The covariant derivative and field strength tensors are defined by 3) where g 1 , g 2 , and g 3 are the gauge couplings of U (1) Y , SU (2) L , and SU (3) C , respectively, and y is the hypercharge. IJK and f abc are the SU (2) L and SU (3) C structure constants, respectively. The 16 baryon and lepton number conserving (∆B = ∆L =0 ) operators involving the field n in SMNEFT are shown in Table 1 [9] in the WCxf convention [13].

Formalism
The anomalous dimensions of an operator are given by the infinite pieces, i.e., the coefficients of the 1/ε terms of the diagrams. In this section, we define our procedure to calculate the ADM, and relegate the details of our calculations to Appendix A. To compute the ADM we use the master formulae presented in Ref. [14]. We compute one-loop contributions to the ADM due to SM gauge couplings. The four-fermion operators (ψ 4 ) in Table 1 can be divided into four categories: (RR)(RR), (LL)(RR), (LR)(RL), and (LR)(LR) on the basis of the chiralities of the fields. The remaining operators are of the form ψ 2 φ 3 , ψ 2 φ 2 D and ψ 2 Xφ. We focus on the ψ 4 -ψ 4 (RR)(RR) and ψ 4 -ψ 2 φ 2 D operator mixing since the mixing between ψ 2 φ 3 , ψ 2 φ 2 D and ψ 2 Xφ has been computed in the Ref. [11] using the background field method. We have checked that the resulting 5 × 5 matrix is consistent with the result for the corresponding SMEFT operators [7] which have a similar ADM structure. For the ψ 4 operators the bare and renormalized operators are related by where the superscript (0) labels the bare matrix elements. Here,Ẑ and Z ψ are the renormalization constants for the operator O and the fields ψ, respectively. In the MS scheme at one-loop level, the renormalization constants take the form, with ψ = {q, u, d, , e} and Z n = 1. The coupling constants are defined by α m = g 2 m /4π with m = 1, 2, 3 for U (1) Y , SU (2) L and SU (3) C , respectively. The coefficients of the UV divergent parts of the diagrams (α m /(4πε)), a m ψ , b m ij and c m ij , are independent of the gauge couplings. Note that c m ij can be related to a m ψ and b m ij via Eq. (3.1). The anomalous dimension matrices are defined where γ C = γ T with γ given by the matrixẐ as and which can be directly expressed in terms of a m ψ and b m ij : Here, the sum is over external fields ψ 1 to ψ 4 in a given operator, and summation over m is implicit. Therefore, in order to compute the ADM for a set of operators, we need to calculate the coefficients a m ψ and b m ij from the field strength renormalization and operator renormalization, respectively.
In Appendix A, we present explicit calculations of the ADMs The same method is applicable to the other operators. It is worth noting that for most of the cases the structure of the ADMs of the SMNEFT operators are similar to those of SMEFT operators [7]. Therefore, our SMNEFT results also serve as an important cross-check for the corresponding gauge terms appearing in the SMEFT ADMs.

Anomalous dimensions
We now present terms for the one-loop ADM that depend on the gauge couplings α 1 , α 2 and α 3 for all 16 SMNEFT operators. The general formula for the ADM is given by Eq. (3.7) and details of the calculations of the Feynman diagrams to extract a m ψ and b m ij can be found in Appendix A. The ADM for bosonic SMNEFT operators is given in Ref. [11]. The ADM of most SMNEFT operators can be obtained from the ADM of the SMEFT operators [7] with a similar structure. equ . We use this procedure as a cross-check when available. No such comparison is possible for O nedu , which has a structure not present in SMEFT.

ψ 4
The ADM for four-fermion operators are provided below.

Operator mixing
We study operator mixing by solving the RG equations presented above in the leading-log approximation. The solution to these equations for running between scales Λ and µ is Depending upon the mixing structure the operators an be divided into five subsets forming 6 × 6, 3 × 3, 3 × 3, 2 × 2, and 2 × 2 ADMs. Defining δC i (µ) = C i (µ) − C i (Λ), the leading-log solution for the first group reads .

(5.2)
Summation over the repeated w index is implicit. Next, we have the 3 × 3 structure, . (5. 3) The operators C nqd and C nqd mix according to .

(5.4)
The operator C n e mix with different flavors: The remaining operators do not mix: To study the running numerically, we set {prst} = {1111} for illustration. We list the 16 × 16 ADM in the basis nqd , C nedu , C n e , C nuq , C φne , C nn } . (5.7) The gauge couplings at 1 TeV are set to g 1 = 0.36, g 2 = 0.64, g 3 = 1.1. The 16 WCs at M Z and at Λ = 1 TeV are related by The running effects in the 6 × 6 and 3 × 3 blocks are small because only electroweak gauge couplings contribute. The mixing in the 2 × 2 block is large as it is governed by QCD.

Phenomenology
We briefly comment on some phenomenological consequences of our results. Semileptonic decays of the b quark are topical given that both charged current and neutral current decay measurements are hinting at new physics. SMNEFT operators lead to the charged current decay b → c n, which contributes at the hadronic level to B → D ( * ) τν τ . They also generate the neutral current decay b → snn which contributes at the hadronic level to B → K ( * ) + invisible decays, which is interpreted as B → K ( * )ν ν in the SM. In the lepton sector, of interest are the FCNC decays τ → µ + invisible and µ → e + invisible. To make contact with low-energy phenomenology, we first run the RG equations down to the weak scale and then match to the low-energy effective field theory extended with right-handed neutrinos n (LNEFT). Depending on the process, further RG running must be performed from the electroweak scale to the appropriate low energy scale such as the m b scale for B meson decay and the m τ scale for τ decay. Note that the sterile neutrino can mix with the active neutrinos, which in itself produces interesting phenomenology, but to keep our discussion simple we neglect this mixing. We select the following four types of process and list the SMNEFT operators relevant to them: nqd , and O nqd can contribute to both the charged current and neutral current decays, and to coherent elastic neutrino-nucleus scattering [12]. For certain flavor combinations, O n e can produce both τ → µ and µ → e decays.
Before studying the low-energy phenomenology, we first run the operators down from the new physics scale Λ to the weak scale µ EW . By using the leading-log approximation in Eq. (5.1), we relate the values of the WCs at M Z to their values at 1 TeV: .

(6.2)
To study the phenomenology at energies below the electroweak scale one can no longer use SMNEFT because of electroweak symmetry breaking. Instead, LNEFT, which respects the SU (3) C × U (1) Q symmetry must be employed to study the processes listed above. We introduce the relevant LNEFT operators and match them with the SMNEFT operators at the weak scale. The SMNEFT operators can generate both neutral and charged current processes after electroweak symmetry breaking. The induced LNEFT operators in the convention of Ref. [10] are displayed in Table 2 and their matching relations at tree level are where we chose a flavor basis in which the left-handed down-type quarks and charged leptons are aligned. The flavor basis for up-type quarks in terms of the mass basis is given by V † u L , where V is the SM CKM matrix. The neutrino fields are in the flavor basis for convenience. In the next subsections, we study the low-energy phenomenology of the listed processes.  Table 2. Here, α is the flavor index of the right-handed neutrino n.  Accounting for QED and QCD running below the weak scale, the one-loop RGE for the four LNEFT operators is given by where e is the QED coupling. Using Eq. (5.1), we relate the four LNEFT operators at the m b and M Z scales: The mixing between O S,RR enud and O T,RR enud is small as it is induced by QED. However, the corresponding mixing of the SMNEFT operators is relatively strong as it comes from electroweak effects. .
The WCs at m b and M Z are related by .
While there is no mixing between the NC LNEFT operators, their corresponding SMNEFT operators can mix above the weak scale. For K → πνν one has to run down to a scale appropriate for kaon decays. . (6.12) The small mixing between these operators is a consequence of QED. For muon decay, one needs to run down to the muon mass.

Electroweak precision observables
The operators O φn and O φne give rise to RH Z-couplings to n and RH W couplings to n and leptons. The RH Z couplings to n can be parameterized in terms of the Wilson coefficient C φn as where g 2 Z = g 2 1 + g 2 2 . Therefore, C φn contributes to the Z-width via Γ(Z → nn). Similary, the RH W couplings can be parameterized in terms of C φne as

Summary
We presented the gauge terms of the one-loop anomalous dimension matrix for the dimension-six operators of SMNEFT; see Eqs. (5.2) to (5.6). We found that renormalization group evolution introduces interesting correlations among observables in different sectors. We discussed a few phenomenological implications of our results. To make contact with low energy observables we also included the matching of SMNEFT to LNEFT at the weak scale and RGE below the weak scale. However, to be confident that cancellations of terms between independent operators are absent, the full one-loop RGE must be calculated.
The UV divergent parts of the current-current diagrams in Fig. 1 mediated by the gauge boson X µ = (B µ , W µ , G µ ) depend on α m and are given by In dimensional regularization, we use the convention d = 4−2ε. Here, D a,b,c represent the symmetric counterparts parts of the diagrams D where J x m (m = 1, 2, 3) are the SU (3) C , SU (2) L and U (1) Y generators. The sum over the index x is implied.
For the penguin insertion in Fig. 2(d), the UV divergent part, if we closeV 1 Γ 1 part of the inserted operator, is given by with the coefficient C d = Tr(V 1 J x m )V 2 ⊗ J x m . Note that, depending upon the structure of the operator given by the matrix (V 1 ) and the type of gauge boson mediated in the penguin diagram Fig. 2(d), the trace can be over the SU (3) C or SU (2) L indices.

A.2 Field strength renormalization
The field strength renormalization constants are defined in Eq. (3.2). At one-loop, these are given by the coefficients where y ψ is the hypercharge of the fields ψ = {q, u, d, , e}.

A.3 Operator renormalization
For illustration, we present an explicit computation of the renormalization constants for O nd , O nedu , O n e and O φn . For the other operators, a similar procedure can be followed. Here we present γ, and in section 4 we present γ C = γ T .
To extract the divergent pieces of the diagrams we use the master formulae of Appendix A.1. For the insertion of O nd = (n p γ µ n r )(d s γ µ d t ) to generate the same, we have (A.11) In this case, D a , i.e., the symmetric counterpart of the first topology in Fig. 1 with X µ = G µ or B µ connected between two d-quark legs is generated. Using Eq. (A.1), the divergent parts of these two contributions are given by Now, using the Eq. (A.7), we obtain As an example of fermionic and bosonic oprator mixing, we present the mixing between O nd and O φn , which is given by Fig. 2(e). Its divergent part reads The operator O n e = (¯ j p n r ) jk (¯ k s e t ) mixes with itself through its insertion into the diagrams D