How dark is the νR-philic dark photon?

We consider a generic dark photon that arises from a hidden U(1) gauge symmetry imposed on right-handed neutrinos (νR). Such a νR-philic dark photon is naturally dark due to the absence of tree-level couplings to normal matter. However, loop-induced couplings to charged leptons and quarks are inevitable, provided that νR mix with left-handed neutrinos via Dirac mass terms. We investigate the loop-induced couplings and find that the νR-philic dark photon is not inaccessibly dark, which could be of potential importance to future dark photon searches at SHiP, FASER, Belle-II, LHC 14 TeV, etc.


Introduction
Right-handed neutrinos (ν R ), albeit not included in the Standard Model (SM), are a highly motivated dark sector extension to accommodate neutrino masses [1][2][3][4][5], dark matter [6][7][8], and baryon asymmetry of the universe [9]. Being intrinsically dark, ν R might have abundant new interactions well hidden from experimental searches. In particular, it is tempting to consider the possibility that there might be a hidden gauge symmetry in the ν R sector [10][11][12][13][14][15][16][17]. The new gauge boson arising from this symmetry does not directly couple to other fermions except for ν R and naturally becomes a dark photon, which we referred to as the ν R -philic dark photon.
The ν R -philic dark photon is not completely dark. It may interact with normal matter via kinetic mixing [18], provided that the new gauge symmetry is Abelian; or, in the JHEP04(2021)003 presence of mass terms connecting ν R and left-handed neutrinos ν L , via one-loop diagrams containing W ± /Z and neutrinos [19]. In the former case, the strength of dark photon interactions with quarks or charged leptons depends on the kinetic mixing parameter in L ⊃ 2 F µν F µν where F µν and F µν are the gauge field tensors of the SM hypercharge U(1) Y and the new U(1), respectively. This case, being essentially independent of the neutrino sector, has been widely considered in a plethora of dark photon studies -for a review, see [20][21][22][23]. In the latter case, the loop-induced couplings depends on neutrino masses and mixing, and will be investigated in this work.
The aim of this work is to address the question of how dark the ν R -philic dark photon could be in the regime that dark-photon-matter interactions dominantly arise from ν L -ν R mixing instead of kinetic mixing. We note here that the dominance might be merely due to accidentally small , or due to fundamental reasons such as the SM U(1) Y being part of a unified gauge symmetry [e.g. SU (5)] in grand unified theories. We opt for a maximally model-independent framework in which generic Dirac and Majorana mass terms are assumed. The loop-induced couplings are UV finite as a consequence of the orthogonality between SM gauge-neutrino couplings and the new ones. Compared to our previous study on loop-induced ν R -philic scalar interactions [24], we find that the couplings in the vector case are not suppressed by light neutrino masses, and might be of potential importance to ongoing/upcoming collider and beam dump searches for dark photons.
The paper is organized as follows: in section 2, we describe the relevant Lagrangian used in this work, reformulate neutrino interactions in the mass basis, and discuss generalized matrix identities for UV divergence cancellation for later use. In section 3, we first derive model-independent expression for effective coupling of Z to charged leptons/quarks through one-loop diagram involving Z and W bosons, respectively. We then evaluate the coupling strength in three different examples. In section 4, we present a qualitative discussion about possible connections between the U(1) R gauge coupling and the mass of Z . In section 5, we present constraints from a vast array of current and future experiments spanning from collider searches to astrophysical phenomena. We finally conclude in section 6 with details of one-loop diagram calculations relegated to appendix A.

Framework
We consider a hidden U(1) gauge symmetry, denoted by U(1) R , imposed on n right-handed neutrinos. The gauge boson of U(1) R in this work is denoted by Z . The relevant part of the Lagrangian for the U(1) R extension reads: 1

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where σ ≡ (1, − σ) with σ being three Pauli matrices; α denotes flavor indices; (i, j) = 1, 2, 3, · · · , n; and Here g R is the gauge coupling constant of U(1) R and Q R,j is the charge of ν R,j under U(1) R . Note that for most general forms of M R and m D , both the Majorana and Dirac mass terms in eq. (2.1) break the U(1) R symmetry. In addition, for arbitrary charge assignments of ν R,j under U(1) R , the model would not be anomaly free. Nevertheless, one can construct complete models in which M R and m D arise from spontaneous symmetry breaking and the cancellation of anomalies can be obtained when several ν R,j 's have different charges with j Q 3 R,j = 0 -see the example in section 3.2. In this section we neglect these model-dependent details and focus on the general framework proposed in eq. (2.1).
The Dirac and Majorana neutrino mass terms in eq. (2.1) can be framed as where ν L = (ν L,e , ν L,µ , ν L,τ ) T and ν R = (ν R,1 , ν R,2 , · · · ) T are column vectors. The entire mass matrix of ν L and ν R can be diagonalized by a unitary matrix U : Here ν i (i = 1, 2, · · · , n + 3) denote neutrino mass eigenstates, with m i being the corresponding masses. We refer to the basis after the U transformation as the chiral basis, and the one before the transformation as the mass basis.
In order to facilitate loop calculations, we need to transform neutrino interaction terms from the chiral basis to the mass basis. In the chiral basis, we have the following neutrino interaction terms: where the first three terms are the SM charged and neutral current interactions, and L denotes left-handed charged leptons. Therefore, in the mass basis, after performing the basis transformation, we obtain: where Here Q R = diag(Q R,1 , Q R,2 , · · · ), I 3×3 is an identity matrix, and 0 x×y is a zero matrix. Notice that some products of the above matrices are zero:

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(2.10) The above results, which will be used in our loop calculations to cancel UV divergences, have been previously derived in ref. [19].

Loop-induced couplings of Z
At tree level, the ν R -philic Z does not directly couple to charged leptons or quarks. At the one-loop level, there are loop-induced couplings of Z generated by the diagrams shown in figure 1.
In the upper and lower panels, we present diagrams in the mass and chiral bases, respectively. The two descriptions are physically equivalent. The diagrams in the chiral basis imply that the loop-induced couplings are proportional to m 2 D , due to the two necessary mass insertions on the neutrino lines. Although in the mass basis this conclusion is not evident, technically our calculations are performed using the diagrams in the upper panel because of properly defined propagators.
Throughout this work, we work in the unitarity gauge so that diagrams involving Goldstone bosons can be disregarded. The detailed calculations are presented in appendix A. The result for a single W ± diagram with neutrino mass eigenstates ν i and ν j running in where u(p 1 ) and u(p 2 ) denote the two external fermion states, µ (q) is the polarization vector of Z µ , and We have adopted dimensional regularization in the loop calculation so the loop integral takes the generalized measure which defines µ and in eq. (3.2).
Note that for each single diagram in the mass basis, the result is UV divergent. However, when we sum over i and j, the UV divergence cancels out. This can be seen as follows: where M 2 d ≡ diag(m 2 1 , m 2 2 , m 2 3 , · · · ) and in the second step we have used eq. (2.10). Eq. (3.3) implies that we can safely ignore the second line in eq. (3.2), as long as eq. (2.10) holds. For a similar reason (G W G † R G † W = 0), the constant term 3 2 can also be ignored. For the Z diagram, we have a similar amplitude for each single diagram. In the softscattering limit (q → 0), we find Once again, we can see that the UV part cancels out during the summation of i and j because ij Hence only the first term in eq. (3.5) needs to be taken into account.

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Summing over i and j in eq. (3.1), we obtain the following effective coupling generated by the loop diagrams: where (3.9)

Example A: 1 ν L + 1 ν R
First, let us consider the simplest case that there are only one ν L and one ν R . The neutrino mass matrix M ν for the case can be diagonalized by a 2 × 2 unitary matrix This unitary matrix can be parametrized as follows where c θ = cos θ and s θ = sin θ. Substituting the explicit form of U in eqs. (2.7) and (2.8), we obtain (3.12) We can now perform the summation in eqs. (3.8)-(3.9). Expanding the result as a Taylor series in s θ (assuming s θ 1) and only retaining the dominant contribution, we obtain where , we can rewrite eqs. (3.14)-(3.15) as JHEP04(2021)003

Example B: 1 ν L + 2 ν R with opposite charges
In this example, we construct a UV-complete model with one ν L and two ν R which have opposite U(1) R charges so that the model is anomaly free. The off-diagonal Majorana mass term does not violate the U(1) R symmetry and the Dirac mass term is generated by a new Higgs doublet H that is charged under U(1) R . The U(1) R charges are assigned as follows: which leads to the following terms that fully respect the U(1) R symmetry: where H ≡ iσ 2 (H ) * . After spontaneous symmetry breaking, H acquires a vacuum expectation value: The neutrino mass matrix for this case can be diagonalized by a 3 × 3 unitary matrix: The texture of the mass matrix on the left-hand side of eq. (3.23) leads to m 1 = 0 and m 4 = m 5 , which is evident from its vanishing trace and determinant. This feature has been often considered in the literature on ν R signals at the LHC -see e.g. [25,26] and references therein. The 3 × 3 unitary matrix can be parametrized as follows Using this form of U in eqs. (2.7) and (2.8), we obtain We can now perform the summation in eqs. (3.8)-(3.9). Expanding the result as a Taylor series in s θ (assuming s θ 1) and only retaining the dominant contribution, we obtain Expressing the results in terms of G F and assuming s θ 1, we obtain We comment here that the above UV-complete and anomaly-free model built on 1 ν L +2 ν R can be straightforwardly generalized to 3 ν L +2n ν R where half of the right-handed neutrinos have opposite U(1) R charges to the other half. Such a generalization can accommodate the realistic three-neutrino mixing measured in neutrino oscillation experiments.

Example C: 3 ν L + 3 ν R with diagonal M R
The most general case with three ν L and an arbitrary number of ν R is complicated and often impossible to be computed analytically. Here we consider an analytically calculable example with 3 ν L + 3 ν R and the following form of the neutrino mass matrix: is not the most general form, but at least it can accommodate the realistic low-energy neutrino mixing responsible for neutrino oscillation.
The mass matrix in this case can be diagonalized by a 6 × 6 unitary matrix: where (s θi , c θi ) ≡ (sin θ i , cos θ i ) and The unitary matrix U can be parametrized as follows Thus, the final unitary matrix U that diagonalizes the original mass matrix is given by

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Substituting it in eqs. (2.7) and (2.8), we obtain Next, we perform the summation in eqs. (3.8)-(3.9), expand the result in s θi , and retain the dominant contribution. The final result reads In the approximation that the ν L -ν R mixing is small, the 3 × 3 unitary matrix U L is almost identical to the PMNS matrix. Due to the presence of off-diagonal entries in U L , g αβ eff,W is generally not flavor diagonal and might lead to observable lepton flavor violation, which will be discussed in section 5.

Dark photon masses and technical naturalness
In this section, we argue that despite being a free parameter, the mass of the ν R -philic dark photon m Z is potentially related to the gauge coupling according to 't Hooft's technical naturalness [27]. Generally speaking, from the consideration of model building and the stability of m Z under loop corrections, we expect that m Z is related to g R by where Λ breaking stands for the symmetry breaking scale of U(1) R . Although without UV completeness we cannot have a more specific interpretation of eq. (4.1), we would like to discuss a few examples to show how m Z is related to g R . First, let us consider that both m Z and M R arise from a scalar singlet φ charged under U(1) R with φ = v R = 0. This leads to m Z ∼ g R v R and M R ∼ y R v R where y R is the Yukawa coupling of φ to ν R . In this case, we consider v R as the symmetry breaking scale Λ breaking so the tree-level relation m Z ∼ g R v R is compatible with eq. (4.1). The Yukawa coupling has an upper bound from perturbativity, y R 4π, which implies that In the absence of a specific symmetry breaking mechanism, we can also obtain eq. (4.2) purely from loop corrections to m Z . If M R breaks the U(1) R symmetry, the Z -Z vacuum Therefore, to make the theory technically natural, the physical mass should not be lower than the loop correction.

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Note, however, that eq. (4.2) is based on the assumption that M R breaks the U(1) R symmetry. If all the Majorana mass terms fully respect U(1) R , such as Example B in section 3, then the symmetry breaking scale can be lower, e.g., determined by m D . Indeed, for the UV complete model in Example B, the symmetry breaking scale is determined by the VEV of the new Higgs doublet H so at tree level we have m Z ∼ g R H and m D ∼ y D H . Then using the perturbativity bound on y D , we obtain Finally, we comment on the possible mass correction from Z-Z mixing. According to the calculation in appendix A, the vacuum polarization diagram leads to mass mixing between Z and Z : where m Z 0 and m Z 0 denote tree-level masses and Here m 2 X causes Z − Z mixing and the mixing angle is roughly | , which must be small. Otherwise, the SM neutral current would be significantly modified and become inconsistent with electroweak precision data. Taking the approximation m 2 Hence we conclude that the mass correction from Z-Z mixing is where we have neglected some O(1) quantities. This mass correction is generally smaller than the right-hand side of eq. (4.3) because m D cannot be much above the electroweak scale.
To summarize, here we draw a less model-dependent conclusion that without finetuning, the ν R -philic dark photon mass is expected to be above the lower bound in eq. (4.2) or eq. (4.3), depending on whether M R breaks the U(1) R symmetry or not, respectively.

Phenomenology
In the previous two sections, we have derived the loop-induced couplings and also argued that from technical naturalness there is a lower bound on the dark photon mass. The results indicate the theoretically favored regime of the mass and the couplings. Therefore, to address the question of how dark the ν R -philic dark photon would be, we shall inspect whether and to what extent the theoretically favored regime could be probed by current and future experiments. In our model, there are effective couplings to both leptons and quarks with comparable strengths. So the experimental constraints on this model are very similar to those on the B − L model. 2 Below we discuss a variety of known bounds that could be important for the ν R -philic dark photon. An overview of existing bounds is presented in figure 2, and the prospect of upcoming experiments in figure 3.

Collider searches
With effective couplings to electrons and quarks, dark photons could be produced directly in e + e − (BaBar, LEP) and hadron colliders (LHC), typically manifesting themselves as resonances in collider signals. For more important. A dedicated analysis on LHC and EWPT bounds and future prospects can be found in ref. [30]. For m Z below the Z pole but above 10 GeV, according to the analyses in [22], the most stringent constraint comes from LHCb di-muon (Z → µ + µ − ) measurements [31]. Below 10 GeV, the BaBar experiment [32] provides more stringent constraints via e + e − → γZ where Z may or may not decay to visible final states. In figures 2 and 3, we present all aforementioned constraints (for compactness in figure 2 they are labeled together as the collider bound). Besides, there is also an indirect LEP bound on four-fermion effective interactions -see section 3.5.2 in ref. [33]. We find that this bound approximately corresponds to g eff /m Z (4.4 TeV) −1 , which is weaker than the aforementioned collider bounds and hence not shown in figures 2 and 3.

Beam dump and neutrino scattering bounds
For 1 MeV m Z 100 MeV, beam dump (BD) and neutrino scattering experiments become important. BD experiments search for dark photons by scattering an electron/proton beam on fixed targets and looking for dark particles that might be produced and subsequently decay after the shield to visible particles such as electrons. A compilation of existing BD bounds from SLAC E141, SLAC E137, Fermilab E774, Orsay, and KEK experiments can be found in [34]. Note that these BD bounds relies on Z → e + e − decay, which implies that such bounds do not apply for m Z 2m e . Nonetheless, below 1 MeV there are much stronger bounds from cosmological and astrophysical observations hence for simplicity we

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do not show the invalidity of BD bounds below 1 MeV. The combined BD bound adopted in this work is taken from [29].
The dark photon in our model could contribute to elastic neutrino scattering by a new neutral-current-like process. Current data from elastic neutrino-electron (CHARM-II [35,36], TEXONO [37], GEMMA [38], Borexino [39], etc.) and neutrino-nucleus (CO-HERENT [40]) scattering are all well consistent with the SM predictions. By comparing the results in refs. [41][42][43], we find that the COHERENT bound is weaker than ν + e scattering bounds, among which the most stringent ones come from CHARM-II, TEXONO, and GEMMA. So the combined result from these experiments is taken from ref. [43] and presented in figures 2 and 3. The future DUNE experiment will be able to further improve the measurement of elastic neutrino scattering [44]. We adopt the DUNE sensitivity from ref. [45] and present it in figure 3.

Astrophysical and cosmological bounds
Astrophysical bounds on dark photons are usually derived from energy loss in celestial bodies such as the sun, red giants, horizontal branch stars, and supernovae. Dark photons may contribute to stellar energy loss directly via dark photon free streaming or indirectly via neutrino production. The enhanced energy loss rate could alter stellar evolution on the horizontal branch in the Hertzsprung-Russell diagram. This sets the strongest limit for sub-MeV dark photons [46]. For smaller m Z , there are also similar bounds from the sun and red giants [46]. We adopt a combined bound from ref. [28] with energy loss via neutrinos taken into account, and refer to it as the stellar cooling bound in figure 2.
The observation of SN1987A can be used to set strong limits on the effective coupling when m Z O(100) MeV [47]. The resulting bound further excludes the space below BD constraints by about three orders of magnitude.
In figure 2 we also show two bounds derived from the effect of Z on big bang nucleosynthesis (BBN). The effect of Z on BBN is two-fold: if Z is light and dominantly decays to invisible states, it would increase the effective number of relativistic dark species N eff . We refer to the bound derived from this effect as the BBN II bound. If Z is heavy, it decays before neutrino decoupling and does not contribute to N eff directly but the neutrino decoupling temperature could be modified if g 2 eff /m 2 Z is comparable to G F (referred to as BBN I). Among various studies on this subject (see e.g. [48][49][50][51][52]), we adopt the bounds from [48] for the B − L model and label them as BBN I and BBN II in figure 2.

Charged lepton flavor violation
The loop-induced couplings do not necessarily conserve lepton flavors, as indicated by eq. (3.40). Note, however, that neither the W -diagram nor the Z-diagram causes flavor violation in the quark sector. In the presence of flavor-changing couplings of Z to charged leptons, there are strong constraints from charged lepton flavor violating (CLFV) decay such as α → β νν, µ → 3e [53], π 0 → eµ; from µ → e conversion in muonic atoms [54], and from the non-observation of muonium-antimuonium transitions [55]. Constraints from α → β γ are weaker since they arise only from two-loop contributions. We do not include CLFV bounds in figures 2 and 3 because such bounds depend on the flavor structure of JHEP04(2021)003 m D which in the Casas-Ibarra parametrization [56]: where R is a complex orthogonal matrix, depends not only on the PMNS matrix U L but also on the R matrix. The effective flavor-changing couplings in the presence of non-trivial R are more complicated and we leave them for future work.

Long-range force searches
Below 0.1 eV, laboratory tests of gravity and gravity-like forces provide highly restrictive constraints, including high precision tests of the inverse-square law (gravity ∝ r −2 ) [57,58] and of the equivalence principle via torsion-balance experiments [59] and lunar laser-ranging (LLR) measurements [59,60]. Besides, measurements of the Casimir effect [61] could set a limit that is slightly stronger than that from the inverse-square law when 0.05 m Z /eV 0.1, which is not presented in figure 2. Also not presented here is the bound from black hole superradiance [62], which would only enter the lower left corner in figure 2. We refer to our previous work [24] for more detailed discussions on the long-range force searches and present only the dominant constraints from torsion-balance tests of the inverse-square law and the equivalence principle. We comment here that neutrino oscillation could also be used to probe long-range forces [63][64][65][66] but similar to the aforementioned CLFV bounds, the flavor structure cannot be simply taken into account by the PMNS matrix. Hence we leave this possibility to future studies.

Prospect of upcoming experiments
Future hadron collider searches could significantly improve the experimental limits on heavy dark photons by almost one order of magnitude, as illustrated in figure 3 by the LHC 14 TeV and future 100 TeV collider sensitivity [30]. Moreover, several LHC-based experiments searching for displaced dark photon decays such as FASER [67], MATHUSLA [68,69], and CodexB [70] will improve the BD bound in the low-mass regime. And the future SHiP experiment [71,72] will substantially broaden the BD bound regarding both the dark photon mass and coupling. The current BaBar bound may be superseded by future bounds from Belle-II [73] and a muon run of NA64 [74,75]. Hence a large part of the space that is often considered for dark photons (20MeV m Z 10 GeV and 10 −8 g eff 10 −3 ) will be probed by future experiments. Here we selectively present the sensitivity curves of SHiP, FASER, NA64µ, and Belle-II. Most of them are taken from ref. [22], except for the FASER/FASER2 sensitivity which is taken from ref. [76].

How dark is the ν R -philic dark photon?
Since the effective coupling g eff is proportional to g R , by tuning down g R one can obtain arbitrarily small g eff to circumvent all constraints presented in figures 2 and 3. On the other hand, if g R is very small, then the lower bounds of m Z discussed in section 4 will also be alleviated, implying that the dark photon could be very light. Taking eqs.

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If the Majorana mass term also breaks the U(1) R symmetry, then the lower bound of m Z is set by eq. (4.2) instead of eq. (4.3). In the standard type I seesaw, we have M R ∼ m 2 D /m ν which implies that for m ν = 0.1 eV and m D = 246 GeV, the U(1) R symmetry breaks at a high energy scale around 10 14 GeV. For this case, we plot the green curve in figure 2. As shown in figure 2, even though with g R 10 −11 the mass of m Z could be below the electroweak scale or lower, the effective coupling is many orders of magnitude below any of known experimental limits.
The inaccessibly large m Z of the green curve is due to the underlying connection between m ν and M R in the standard type I seesaw. In some alternative neutrino mass models such as inverse seesaw [77], the scale of M R is decoupled from m ν , which allows for a sizable ν L -ν R mixing even when M R is reduced to the TeV scale, and has motivated many studies on collider searches for right-handed neutrinos -see ref. [26] for a review. Here for illustration we simply set M R = m D / sin θ with m D = 246 GeV and sin θ = 10 −2 , which ensures that ν R is sufficiently heavy to avoid all current collider bounds. The possibility of collider-accessible ν R involves more complicated phenomenology which is beyond the scope of this work. The strength of g eff and the lower bound of m Z in this case is presented by the orange lines in figures 2 and 3. Now confronting the theoretically favored g eff and m Z of the aforementioned three scenarios with the experimental limits, we can see that only when the U(1) R breaking scale is determined by m D or M R = m D / sin θ with sizable sin θ, the ν R -philic dark photon could be of phenomenological interest. The former could potentially give rise to observable effects in long-range force searches, astrophysical observations, beam dump and collider experiments. The latter, albeit beyond the current collider bounds, might be of importance to future collider searches. In addition, the SHiP experiment will be able to considerably dig into the parameter space of the latter.

Conclusion
The ν R -phillic dark photon Z which arises from a hidden U(1) R gauge symmetry and at the tree-level couples only to the right-handed neutrinos, interacts weakly with SM particles via loop-level processes -see figure 1. Assuming the most general Dirac and Majorana mass matrices, we have derived loop-induced couplings of Z to charged leptons and quarks. The results are given in eqs. (3.8) and (3.9), which are applied to a few examples including a UV complete model. For a special case with three ν L and three ν R , the loop-induced coupling are given by eqs. (3.40) and (3.41). We have also discussed potential connections between the mass m Z and the gauge coupling g R from the point of view of technical naturalness, which implies that m Z should be generally above the lower bound in eq. (4.2) if M R breaks U(1) R , or the bound in eq. (4.3) if only m D breaks the symmetry.
The theoretically favored values of the loop-induced couplings are confronted with experimental constraints and prospects in figures 2 and 3. We find that the magnitude of loop-induced couplings allows current experiments to put noteworthy constraints on it. Future beam dump experiments like SHiP and FASER together with upgraded collider searches will have substantially improved sensitivity on such a dark photon.

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Hence as the answer to the question proposed in the title, we conclude that the ν Rphilic dark photon might not be inaccessibly dark and could be of importance to a variety of experiments!

Acknowledgments
The work of G.C. is supported in part by the US Department of Energy under Grant No. DE-SC0017987 and also in part by the McDonnell Center for the Space Sciences. X.J.X is supported by the "Probing dark matter with neutrinos" ULB-ARC convention and by the F.R.S./FNRS under the Excellence of Science (EoS) project No. 30820817 -be.h "The H boson gateway to physics beyond the Standard Model". We acknowledge the use of Package-X [78], which is a great tool to simplify the loop calculations in this work.

A Explicit calculation of loop diagrams
In this appendix, we compute loop diagrams presented in figure 1 in the mass basis. In the main text, we use two-component Weyl spinors for conceptual simplicity. However, technically it is more convenient to convert them to four-component Dirac/Majorana spinors so that the standard trace technology can be employed. Following the same convention as ref. [24], we rewrite eq. (2.6) as For simplicity, we symbolically denote the relevant product of neutrino-gauge couplings by G X (it may stands for different quantities in different diagrams), which will be replaced by specific couplings when actually used.

A.1 The Z diagram
The diagram is presented in the upper right panel in figure 1. We first compute the vacuum polarization part of the diagram (i.e. without the external fermion lines): where q is the momentum of Z and Taking into account the Lorentz structure of the amplitude, this can be further decomposed as: The full amplitude of the Z diagram can be written as where the most general form of ∆ µν Z (q) in R ξ gauges is We proceed with the unitarity gauge corresponding to ξ → ∞, and the soft-scattering limit q m Z : (A.10) By applying the result of eq. (A.3) to eq. (A.8), we obtain iM Z = −i G X 16π 2 m 2 Z F 1 (m i , m j , q 2 ) q µ q ν + F 2 (m i , m j , q 2 ) g µν u(p 1 )γ ν P L/R u(p 2 ), (A.11) where F 1 and F 2 were already given in eqs. (A.6) and (A.7), respectively.

A.2 The W diagram
The diagram is presented in the upper left panel in figure 1. The amplitude reads: (A.14) Similar to the Z diagram, we take the unitarity gauge (ξ → ∞) and the soft-scattering limit (q → 0). The quantity in the loop integral is proportional to d 4 k (2π) 4 γ ν P L ∆ j (k−p 1 )γ ρ P L ∆ i (p 2 −k)γ µ P L ∆ W µν (k) ≡ C a γ ρ P L +C b P L p ρ 1 +C c P L p ρ 2 . (A.15)

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Here (C a , C b , C c ) are functions of scalar invariants p 2 1 and p 2 2 . The last two terms are suppressed when imposing the on-shell conditions. Focusing only on the γ ρ P L term, we obtain Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.