One-loop corrections to light neutrino masses in gauged U(1) extensions of the standard model

We consider gauged U(1) extensions of the standard model of particle physics with three right-handed sterile neutrinos and a singlet scalar. The neutrinos obtain mass via the type I seesaw mechanism. We compute the one loop corrections to the elements of the tree level mass matrix of the light neutrinos and show explicitly the cancellation of the gauge dependent terms. We present a general formula for the gauge independent, finite one-loop corrections for arbitrary number new U(1) groups, new complex scalars and sterile neutrinos. We estimate the size of the corrections relative to the tree level mass matrix in a particular extension, the super-weak model.

corrections may be sizable and can affect significantly the validity region in the parameter space of the model. For instance, the lightness of active neutrinos requires that the loop corrections to the mass matrix of those particles must also be small in order to have a phenomenologically viable model. Computations of such one-loop corrections have been carried out previously in Refs. [21,22] for the canonical seesaw case, and in the context of multi-Higgs doublet models [23][24][25][26]. In the cases of gauged U(1) models we are not aware of a computation of the one-loop corrections to active neutrino mass matrix.
In this article we consider gauged U(1) extensions of the SM and derive a general formula for the one-loop corrections of the mass matrix of the active neutrinos. The mass matrix of the active neutrinos emerges after SSB due to the type I seesaw mechanism. Our goal is to derive the one-loop corrections to that mass matrix and estimate their sizes relative to the tree level for a particular example, called the super-weak force [27]. The super-weak model contains three additional right-handed sterile (under the SM interactions) neutrinos and one complex scalar field in addition to the fields of the SM. The loop corrections involve all the gauge and scalar bosons which couple to neutrinos.
In order to obtain the one-loop corrections to the elements of the light neutrino mass matrix, we perform our computations in the R ξ gauge and show explicitly the intricate cancellation of the gauge fixing parameters from the corrections. In addition, we shall also demonstrate the cancellation of the poles when the loop integrals are regulated by dimensional regularization in d = 4 − 2 dimensions. These cancellations are highly non-trivial, and therefore provide strong checks on the correctness of the computations.
The paper is composed as follows. We introduce the model to the extent needed for the present work in Sect. II. We define and compute the one loop correction to mass matrix of the active neutrinos in Sect. III. In Sect. IV we provide numerical estimates of the one-loop corrections and show that those are very small. Finally we summarize our findings in Sec. V.
We collect auxiliary formulas in the appendices and also provide an auxiliary zip file containing the SARAH model, parameter and particle files.

II. PARTICLE MODEL, MIXINGS AND INTERACTIONS
We consider an extension of the standard model by a U(1) z gauge group with particle content and charge assignment defined in Ref. [27]. The super-weak model is an economical extension of the standard model that provides a framework to explain the origin of (i) neutrino masses and oscillations [28], (ii) dark matter [29], (iii) cosmic inflation and stabilization of the electroweak vacuum [30], (iv) matter-antimatter asymmetry of the universe. The complete model including Feynman rules in the unitary gauge was presented fully in Ref. [27]. As we are to compute one-loop corrections to neutrino masses, we recall the details relevant to such computations, with Feynman rules in the R ξ gauge. We generated those Feynman rules with SARAH [31][32][33][34] but here we present simpler forms for the rules needed in our computations to make those more comprehensive. We also recall some of the conventions that are different in SARAH and the original definition of the model. We stick to the SARAH conventions throughout this work. [35] A. Mixing of neutral gauge bosons The particle content of the standard model is extended by 3 right-handed neutrinos ν R , a new scalar χ, and the U(1) z gauge boson B . As the field strength tensors of the U(1) gauge groups are gauge invariant, kinetic mixing is allowed between the gauge fields belonging to the hypercharge U(1) y and the new U(1) z gauge symmetries, whose strength is measured by in where B µ is the U(1) y gauge field. However, equivalently, we can choose the basis-the convention in SARAH-in which the gauge-field strengths do not mix, while the couplings are given by a 2 × 2 coupling matrix in the covariant derivative where y and z are the U(1) y and U(1) z charges. We can parametrize the coupling matrix aŝ The coupling mixing matrix containing η is equivalent to the kinetic mixing in the Lagrangian (II.1) and the parameters of the two representations are related by g z = g z / √ 1 − 2 and η = g y /g z . In this paper, it will be convenient to use the kinetic mixing representation defined by (II.1).
The rotation with angle is unphysical as it can be absorbed into the mixing of the neutral gauge fields B µ , B µ and W 3 µ to the mass eigenstates A µ , Z µ and Z µ , which then can be described by a rotation matrix This matrix depends on two mixing angles: θ W is the weak mixing (or Weinberg) angle and θ Z is the Z − Z mixing angle [36]. In terms of the coupling parameters κ = cos θ W (γ y − 2γ z ) and τ = 2 cos θ W γ z tan β , (II. 5) introduced in Ref. [27], this new mixing angle is given implicitly by tan(2θ Z ) = 2κ/(1−κ 2 −τ 2 ).
In Eq. (II.5) tan β = w/v is the ratio of the vacuum expectation values (VEVs) of the scalar fields (see below) and γ y = ( / √ 1 − 2 )(g y /g L ), γ z = g z /g L , i.e. the couplings are normalized by the SU(2) L coupling.
We can express the elements of the Z − Z mixing matrix explicitly, which also appear in the neutral currents where e is the electromagnetic coupling and P R/L ≡ P ± = 1 2 (1 ± γ 5 ) are the usual chiral projections. In particular, for neutrinos i.e. C L/R Z νν can be obtained from C L/R Zνν by the replacement

B. Mixings of scalar and Goldstone bosons
In addition to the usual SU (2) L -doublet Brout-Englert-Higgs (BEH) field there is another complex scalar χ in the model, with charges specified in [27]. The Lagrangian of the scalar fields contains the potential energy where |φ| 2 = |φ + | 2 + |φ 0 | 2 . In the R ξ gauge we parametrize the scalar fields after spontaneous symmetry breaking as where v and w denotes the vacuum expectation values (VEVs) of the fields, whose values are Using the VEVs, we can express the quadratic couplings as The fields h and s are two real scalars and σ φ and σ χ are the corresponding Goldstone bosons that are weak eigenstates. We shall denote the mass eigenstates with h, s and σ Z , σ Z .
These different eigenstates are related by the rotations and where θ S and θ G are the scalar and Goldstone mixing angles that can be determined by the diagonalization of the mass matrix of the real scalars and that of the neutral Goldstone bosons.
The mass matrix of the Goldstone bosons is given in principle by the sum of gaugeindependent and gauge-dependent terms. However, the gauge-independent terms vanish by Eq. (II.13):  so the mass matrix contains only gauge-dependent terms, where ξ Z and ξ Z are the gauge parameters. The mass matrix is symmetric, so we can write it formally as for both x = Z and Z . Explicitly, 20) where M W = vg L 2 is the mass of the W bosons, and the elements of m 2 A Z can be obtained by the replacement Z → Z in the chiral couplings, which implies the replacement (II.8) in the second forms of the matrix elements. The latter are the most convenient ones for the diagonalization of the mass matrix. Using Eq. (II.6), one can check that the matrix is indeed diagonal provided we have for the Goldstone mixing angle and As expected, the elements of the diagonal matrix m 2 diag,A coincide with the squares of the masses of the neutral gauge bosons [27], and which can also be expressed conveniently with the chiral couplings and Goldstone mixing angle.
First we note that using Eq. (II.23), we find the simple relation between the Goldstone and neutral boson mixing angles, and also and also using Eq. (II.8), From Eq. (II.25) and (II.26) we can express which after substitution and simple rearrangement leads to (II.30)

D. Mass terms and mixing of neutrinos
The masses of the neutrinos are generated by the leptonic Yukawa terms in the Lagrangian [27], where L L is the Dirac adjoint of the left handed lepton dublet, Y N and Y ν are 3 × 3 matrices, the superscript c denotes charge conjugation, ν c = −iγ 2 ν * . After SSB this Lagrangian becomes 32) and the terms proportional to the VEVs provide the mass matrices where the Majorana mass matrix M N is real and symmetric, while the Dirac mass matrix M D is complex and Hermitian.
In flavour basis the 6 × 6 mass matrix for the neutrinos that can be written in terms of 3 × 3 blocks as The weak (flavour) eigenstates (ν e , ν µ , ν τ , ν R,1 , ν R,2 , ν R,3 ) can be transformed into the basis of ν i (i = 1 − 6) mass eigenstates with a 6 × 6 unitary matrix [37] U where the mass matrix is diagonal, It is helpful to decompose the matrix U into two 3 × 6 blocks U L and U * R , where both blocks are 6 × 3 matrices. It may be worth to emphasize that in spite of what might be implied by the notation, the matrices U L and U * R are only semi-unitary. Useful relations of these matrices are collected in Appendix A.

E. Gauge boson -neutrino interactions
As the neutral currents are written in terms of flavour eigenstates, the interactions between the neutral gauge bosons and the propagating mass eigenstate neutrinos include also the neutrino mixing matrices: for both V = Z and V = Z .

F. Scalar boson -neutrino and Goldstone boson -neutrino interactions
The terms containing the scalar and Goldstone bosons in Eq. (II.32) provide interactions between those and the neutrinos. These interactions have the same structure with small differences. For the propagating scalar states S k or σ k (k = 1 denoting h or the Goldstone boson belonging to Z and k = 2 referring to s or the Goldstone boson belonging to Z ) such interactions can be decomposed into left and right chiral terms where the matrices Γ L/R contain both the mixing matrix of the neutrinos and the mixing matrix of the scalar or Goldstone bosons. The left chiral coefficients are and the right chiral ones are related by complex conjugation,

III. NEUTRINO MASS MATRIX AT ONE-LOOP ORDER
We are interested in the one-loop correction δM L to the tree-level mass matrix of the light neutrinos. In perturbation theory we deal with propagating states which are mass eigenstates. Hence, the correction is obtained by Using Eq. (II.36), we can compute the 3 × 3 blocks as In the following subsections we prove that the one-loop correction to the mass matrix of the active neutrinos have the form where we introduced the finite matrix valued function of dimension mass and with summation running over all neutrinos.

A. Self-energy decomposition
The neutrino self energy is a 6 × 6 matrix that can be decomposed as Using this decomposition, δM L is given by [23] δM The

B. Contributions with neutral gauge bosons in the loop
The contribution of the neutral gauge boson V is where ξ V is the gauge parameter and . (III.9) Introducing the 6 × 6 matrix The contribution of the neutral Goldstone boson σ V (V = 1 means the Goldstone boson belonging to the Z field and V = 2 refers to the Z field) is (III.12) Using the matrix notation, we can write ( and (A.5), we obtain the correction to the mass matrix as (III.14) We now substitute Mm (1) M = m (3) and using Eq. (II.30), we obtain (III.15)

D. Contributions with scalar bosons in the loop
The scalar -neutrino vertex is very similar to the Goldstone boson neutrino vertex, so the contribution with a scalar boson S k in the loop can be written immediately in analogy with Eq. (III.14): (III.17) Introducing the integral , (III.18) the matrix I (n) with elements and using the relations (II.30) allows us to recast Eq. (III.17) into a neatly condensed form and (III. 27) In order to prove that δM L is finite, one has to show that δM (s) L is free from poles. We prove that it in fact vanishes because the matrix I (s,1) is zero matrix due to the identity (A.5), while the coefficient in the second term cancels because the matrices Z S and Z G are orthogonal, so Hence the mass independent terms, including the divergent pieces of the light neutrino one-loop mass correction cancel, and we can set = 0, which yields δM L = δM  The neutrino mass and mixing matrices with arbitrary n a and n s are written identically in the block form, differing only on the block shape: U L is a n a × (n a + n s ) matrix and U R is a n s ×(n a +n s ) matrix. The finite correction derived in Eq. (III.4) is then immediately generalized where the upper limit in the summation in the matrix F is n a + n s . The factor 3 in front of the first term in the bracket of Eq. (III.30) stems from the three polarization states of the propagating massive neutral gauge bosons. The corresponding factor is of course one in the case of the scalars. This formula is also independent of the new U(1) charge assignments.

IV. NUMERICAL ESTIMATE OF THE CORRECTIONS
We are now ready to estimate the order of magnitude of the corrections. We assume large mixing in the scalar sector: θ S = O(1). The Z mass and mixing angle θ G are fixed by the gauge couplings g y = γ y g L and g z and ratio of VEVs, tan β ≡ w/v. We plot their magnitudes in Fig.   2, scanning the parameters g y , g z ∈ [10 −6 , 1] and for w = 100, 750 GeV. Note that larger tan β corresponds to a larger Goldstone angle. Smaller tan β distorts the M Z contours so that the same Z mass can be achieved with larger gauge couplings g y and g z compared to large tan β.
In addition, we set M s /v = O(1), that is, only the mass of the Z boson is free, and may be far from electroweak scale. The relevant gauge couplings can then be estimated as from Fig.   2 after identifying the region in (g y , g z ) plane corresponding to M Z ∈ [20,200] MeV, which is the relevant mass region for the super-weak model to reproduce the dark matter relic density, allowed by experimental constraints [29]. Then we identify the order-of-magnitude estimate for | sin θ G | by comparing the regions relevant to the mass range of M Z . For w = 100 GeV, we have | sin θ G | < 10 −6 , which we take as a conservative upper limit. Then the prefactors in gauge boson contributions to δM L are and (IV.2) Then the numerical estimate for the total correction in Eq. (III.4) can be written as  active and sterile neutrinos [28].

Acknowledgments
We are grateful to Josu Hernández-García for discussions on this project. This work was supported by grant K 125105 of the National Research, Development and Innovation Fund in Hungary.
Appendix A: Some properties of neutrino mass and mixing matrices In this appendix, we derive some useful relations among the neutrino mass and mixing matrices. The matrix U that diagonalizes the neutrino mass matrix is unitary, hence from which we obtain the following important relations: and where 1 n denotes the n × n unit matrix. The second unitarity conditions gives Using Eq. (II.35) we derive where the last term vanishes by Eq. (A.5), so Finally, Expanding the factors into the parenthesis, the first two terms give vanishing contribution by where the second term on the right does not vanish this time.

Appendix B: Evaluation of the vector boson exchange diagram
The vector boson exchange diagrams, shown in the bottom row of Fig. 1  Ref. [40], Eq (4.4) contains the self energy: where the 6 × 6 matrices are defined as follows. The matrix P is the fermion propagator, diagonal in the mass eigenstates, In the following we shall write P for P(p − ). The matrix A is self-adjoint, A † = A, and so is Γ L/R . We also introduce the abbreviatioñ which will simplify our calculations. In order to compute the loop integral easily containing the neutral vector boson propagator in the neutrino self-energy loop, in this Appendix we perform tensor reduction of the matrix product such that the numerator factor be at most linear in the loop momentum .
When the fermion momentum p appears as The chiral coupling matrix A anticommutes with the Dirac matrices γ µ , hence and similarly, Multiplying Eqs. (B.6) and (B.7), we obtain the expression (B.4), and its expansion yields Using that ¡ pA =Ã ¡ p, the fourth term can rearranged as The ¡ p is on extreme left and right, hence can be replaced with M, giving Our goal is to compute the one-loop correction (III.7) to the tree-level mass matrix of the light neutrinos. In order to obtain it, one sandwiches the left handed pieces M L i between the matrices U * L and U † L . Using the properties of the neutrino mixing matrices of Appendix A, we immediately see that Then using the matrix relations derived in Appendix A, we can compute the following identities: Finally sandwiching Eq. (B.20) gives us As mentioned, the last term is proportional to ( / p − / ), but only the term with / contributes to B L (p = 0). That piece, being an odd function of , vanishes upon integration, which completes the proof of Eq. (B.16).
The charged vector bosons W ± also contribute to the neutrino self-energy. The corresponding Feynman rules are with U L being the charged lepton mixing matrix and Γ L W −¯ ν = (Γ L W + ν ) † . The charged vector boson contribution to (III.7) is proportional to U * L MU † L , which vanishes identically as shown in Appendix A.