Neutrino-Electron Scattering: General Constraints on Z' and Dark Photon Models

We study the framework of $U(1)_X$ models with kinetic mixing and/or mass mixing terms. We give general and exact analytic formulas and derive limits on a variety of $U(1)_X$ models that induce new physics contributions to neutrino-electron scattering, taking into account interference between the new physics and Standard Model contributions. Data from TEXONO, CHARM-II and GEMMA are analyzed and shown to be complementary to each other to provide the most restrictive bounds on masses of the new vector bosons. In particular, we demonstrate the validity of our results to dark photon-like as well as light $Z^\prime$ models.


Introduction
The Standard Model provides an elegant and successful explanation to the electroweak and strong interactions in nature [1]. However, there are many open problems that require physics beyond the Standard Model (SM) to take place. A common particle in these models beyond the SM is a neutral gauge boson, usually referred to as Z . Such gauge bosons arise naturally in Abelian gauge groups or in gauge groups that embed an Abelian symmetry. The mass of this boson can be generated in several ways, for instance via spontaneous symmetry breaking of a scalar which is singlet or doublet under the SM group, each case leading to very different phenomenology [2]. The mass and interaction strength of Z bosons with SM particles are very model dependent and entitled to a rich phenomenology from low to high energy scales [3,4]. Phenomenological studies have been conducted, among others, in the context of colliders [5][6][7][8][9][10][11][12][13][14][15][16][17][18], electroweak precision [19,20], flavor physics [21][22][23] or neutrinoless double beta decay [24].
Another particle often present in a multitude of beyond the Standard Model frameworks is the dark photon. Dark photons are typically defined as light vector bosons that possess small kinetic mixing with the QED field strength tensor [25][26][27][28]. They are supposed to be much lighter than 90 GeV, the mass of the Z boson. Such particles have also been subject of intense searches at low energy colliders and accelerators [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]. We emphasize that when we refer to the Z mass in our work, we mean the gauge boson mass in a general way, because the Z can take the form of a dark photon-like boson. The main difference between these two bosons is the type of interactions they feature with the SM particles. If the kinetic mixing with the QED field strength tensor was the only new term, then the dark photon would only have vectorial interactions with quarks and charged leptons and none whatsoever with the SM neutrinos. In complete dark photon-like models, however, there should be an underlying new broken gauge symmetry, under which SM particles possibly have non-zero charges. Therefore, there could be a mass mixing term in addition to the kinetic mixing, and we arrive again at the classical notion of a Z boson. For historical reason, Z models and dark photon models are usually described in different contexts but they simply refer to a massive gauge boson coming from a new gauge symmetry. Therefore, in what follows, we will use the terms Z and dark photon interchangeably.
Phenomenological studies in the context of Z or dark photon models in a general setup should include the presence of both mass and kinetic mixing between the vector bosons in the theory. In this work, we provide a general formalism to treat these models. In what regards phenomenology, we will be focused primarily on neutrino-electron scattering process [48][49][50][51][52][53][54][55] since both Z as well as dark photon-like models give rise to sizable new physics contributions, allowing us to place restrictive constraints.
The observation of neutrino-electron scattering has proven to be an amazing laboratory to test the SM and probe new physics effects motivating a multitude of studies [56][57][58][59][60][61][62][63]. In particular, precise measurements of the neutrino-electron scattering have furnished relevant bounds on Z bosons for specific models based on the baryon minus lepton number (B − L) [64], L µ − L τ symmetries [65][66][67] and dark photon-like models [68,69]. In the future, more measurements will be coming up, see e.g. Refs. [70,71]. Motivated by the popularity of Z and dark photon models in the literature and the relevance of neutrino-electron scattering constraints for light dark species we build here up a general setting where constraints using data from neutrino-electron scattering can be placed on Z and dark photon models in the presence of mass and kinetic mixing terms.
The paper is build up as follows: In Section 2 we develop the general formalism to describe kinetic and mass mixing with of a general Z with the SM. Exact analytical expressions are provided. Section 3 derives the interactions relevant for neutrino-electron scattering and gives expressions for cross sections. The fitting procedure is described in Sec. 4, bounds on the masses and couplings from TEXONO, CHARM-II and GEMMA data are discussed in Section 5, before we conclude in Section 6. Various technical details and lengthy analytical expressions are delegated to appendices.

General U (1) X Models
The formalism for Z − Z mixing has been frequently discussed in the literature, see e.g. [72] 1 . Here we develop the framework in our notation and give exact expressions without any approximation.
In the presence of the gauge groups SU (2) L × U (1) Y × U (1) X , the gauge bosons are denoted aŝ W a (a = 1, 2, 3),B andX, respectively. The Lagrangian can be written as Here f = (ν L , e L ) T , e R , ν R , . . . stands for all chiral fermions in the model and φ stands for all scalar bosons. Since we are considering the most general case, the fermion and scalar contents are not necessarily the same as in the SM. For example, f may include right-handed neutrinos ν R ; the scalar sector may contain more than one Higgs doublet or singlets. The covariant derivative is given by where t a = σ a /2 are the generators of SU (2), g X is the gauge coupling of the new U (1) X and Q X,Y are the operators projecting the charges of the particles under U (1) X and U (1) Y . The two U (1) gauge bosonsB andX couple to each other via the term 2B µνX µν , which induces kinetic mixing of X with the other gauge bosons. This term is essentially guaranteed since it is generated at loop-level even if zero at some scale [74], if there are particles charged under hypercharge and U (1) X . The mass mixing requires that there is a scalar that is charged under the SM and the U (1) X groups. It is thus absent if the U (1) X is broken by SM singlet scalars. The fact that the U (1) X and the SU (2) L × U (1) Y are broken gauge symmetries leads to mass terms for the gauge bosons as well to mass mixing terms. Appendix A shows the structure of those terms for an arbitrary number of singlet and doublet fields in the realistic scenario in which U (1) em remains unbroken, see Eqs. (A.11) and (2.9).
We would like to comment here that in the exact physical basis where all gauge bosons have canonical kinetic terms and are mass eigenstates, the mixings mentioned above will be completely removed and converted to corrections to the gauge-fermion or gauge-scalar interactions. Next, we shall show this explicitly. The equations in this section are exact, without any approximation such as 1 or g X 1.
We define three bases for the corresponding gauge bosons: • Fundamental basis, which is defined as the basis we start with in Eq. (2.1); gauge bosons are denoted asX • Intermediate basis, where the gauge bosons have canonical kinetic terms but a non-diagonal mass matrix, denoted as • Physical basis, where gauge bosons are mass eigenstates with canonical kinetic terms, denoted The three bases can be transformed to each other by where L is a non-unitary linear transformation while U is a unitary transformation, which will be derived below.
First, the kinetic terms in the first row of Eq. (2.1) can be regarded as quadratic-form functions ofX (here we can ignore the cubic and quartic terms of non-Abelian gauge bosons), In the second step we have diagonalized the matrix by L , given as follows 3 : Eq. (2.7) implies that after the transformationX → X = L T X , the kinetic terms become canonical.
Next we shall diagonalize the mass matrix of gauge bosons. Although the mass matrix depends on details of the scalar sector, such as the numbers or types of new scalars introduced, we show in the appendix that as long as these scalars do not break the U (1) em symmetry, the mass matrix in the fundamental basis is always block diagonal of the form diag(m 2 W , m 2 W ,M 3×3 ). Moreover,M 3×3 can be further block-diagonalized intoM Although (W 1 , W 2 ) are mass eigenstates with the same mass m 2 W , in the SM they are conventionally converted to W ± = 1 √ 2 (W 1 ∓ iW 2 ). In this paper, the (W 1 , W 2 ) or W ± sector is exactly the same as in the SM. When discussing the bases, we still use (W 1 , W 2 ) for simplicity; later in the charged-current interactions we use W ± . The conversion is the same as in the SM. 3 Note that L is not unique, e.g. one can also use Actually this transformation can be any matrix of the form L O 5 where O 5 is a 5 × 5 orthogonal matrix.
by the Weinberg rotation where s W = sin θ W , c W = cos θ W . In the above expression forM 3×3 , the parameters z, δ and x are related to mass mixing and exact formulas are given in Appendix A. In the intermediate basis, according to Eq. (2.6), the mass matrix is A useful result which can be verified by simple calculation is that the product M 3×3 (s W , c W , 0) T is zero. It implies that (s W , c W , 0) T is one of the eigenvectors of M 3×3 , which significantly simplifies the diagonalisation process of M 3×3 . The other two eigenvectors should be orthogonal to this one and can be parametrized by an angle α. So all three eigenvectors are given by the columns of the matrix where s α = sin α and c α = cos α. One can use U to diagonalize M 3×3 : , (2.14) which has the following limit if → 0 and δ → 0: Eq. (2.13) implies that the transformation from the intermediate basis to the physical basis is given by X → Z = U T X.
In summary, the gauge bosons mass terms in the three bases are given by (2.16) Now that we have the transformations between the three bases, we are ready to derive the gaugefermion interactions in the physical basis. Note that the transformations represented by L and U in Eqs. (2.8) and (2.12) are only limited to the lower 3 × 3 block. Therefore the charged current interaction mediated by the W ± bosons is the same as in the SM. We only need to consider the interactions of fermions with the remaining three gauge bosons, namelyŴ 3 µ ,B µ , andX µ in the fundamental basis or A, Z, and Z in the physical basis. Using Eq. (2.2) these interactions are obtained from Here t 3 , Q Y and Q X depend on the representation of f in the gauge groups. To proceed with the analysis, we change somewhat the notation and disassemble the SU (2) doublets of fermions and regard t 3 as a quantum number rather than a Pauli matrix. For example, when f is ν L or e L , t 3 takes the values 1/2 or −1/2, respectively. In this sense, 2 ) can be treated as a vector of numbers rather than 2 × 2 matrices. Therefore, the interactions in the basis of A, Z, and Z can be derived by where J µ em and J µ NC are the electromagnetic and neutral currents in the SM respectively, and J µ X is a new current which we will refer to as the X-current. For the convenience of later use, we explicitly write them down: With these currents at hand, some comments are in order: • In the limit α → 0, the SM is recovered.
• α contains not only the kinetic mixing but also the mass mixing δ in Eq. (2.9).
• The charged current and electromagnetic interactions are the same as in the SM.
The currents allow one to calculate physical processes, which is what will be done in the next section.
3 Neutrino-Electron Scattering in U (1) X Models As we discussed in the previous section, neither the kinetic mixing nor the mass mixing of the gauge bosons change the charged current and electromagnetic interactions. Therefore, neutrinos still have the SM charged current with electrons and are not involved in electromagnetic interactions [64,68]. In addition to the charged current, neutrinos can interact with electrons via Z µ and Z µ . We re-emphasize that generally we refer to dark photon-like and Z gauge bosons as simply Z gauge bosons. The relevant interactions for neutrino-electron scattering can be written as follows: (3.8) Table 1. Anomaly free charge assignments in U (1) X models [75]. Those were designed to explain neutrino masses via the seesaw mechanism and address the flavor problem in two Higgs doublet models.
We note that even though Q R Xν appears in the above interactions, it will disappear in the cross sections of neutrino-electron scattering because for realistic neutrino sources, neutrinos are always produced via the charged current, which means the sources only emit left-handed neutrinos (or righthanded antineutrinos). From Eq. (3.1), one can see that left-handed neutrinos can not be converted to right-handed neutrinos in any of the vertices. Therefore right-handed neutrinos are irrelevant to neutrino-electron scattering.
So far the discussion was general. In explicit UV-complete and self-consistent models the U (1) X charges should take specific values to guarantee anomaly cancellation. In this work, we will adopt the anomaly free charge assignments as listed in Tab 1. The models are adopted from Ref. [75], where they were studied in the context of two Higgs doublet models without tree-level flavor changing neutral currents. They are characteristic for many of the available U (1) X models in the literature, and can serve as benchmark models for out study.
We compute the cross sections of neutrino-electron scattering in the appendix. The results are: where E ν and T are the neutrino energy and electron recoil energy respectively, and g 1, 2 are two dimensionless quantities with quite complicated expressions. They can be decomposed into several parts, to be discussed below. Eq. (3.9) or Eq. (3.10) assume that the initial neutrinos are left-handed neutrinos or right-handed antineutrinos, respectively. As one may notice, the difference between the two cross sections is simply an interchange between g 1 and g 2 , Eqs. (3.9) and (3.10) can be used for both electron neutrinos and muon neutrinos. For electron neutrinos, additional charged current contributions should be taken into account, which changes the SM part of g 1 and g 2 .
Next we discuss the two dimensionless quantities g 1, 2 . They can be decomposed into three parts, referred to as the SM part g SM 1, 2 , the new Z µ -mediated part a 1, 2 (see below), and the Z µ -mediated part b 1, 2 r: The explicit expressions of g SM 1, 2 depend on whether the charged current contributions should be taken into account: The Z µ -mediated part b 1, 2 r can change drastically for very light Z µ , this is quantified by the parameter r defined as Hence, if we observe a potential Z signal much below m min Z , the overall effect will be independent of its mass.
We shall now turn our attention to the quantities a 1, 2 and b 1, 2 . They are small if the gauge coupling g X , the kinetic mixing and the mass mixing s α are small as well, i.e. (3.16) Without assuming any of them to be small, the exact expressions are computed in the appendix and summarized below: (3.20) The interpretation of those terms is straightforward. The terms a 1,2 contain contributions proportional to sin 2 α and sin α cos α. Those take into account the mixing of the SM Z with the new Z , and would disappear of the mass and kinetic mixing terms would both vanish, = δ = 0, see Eq.
(2.14). The would-be SM Z boson has an admixture of the new gauge bosons and its coupling with neutrinos and electrons is modified accordingly. In analogy the terms b 1,2 contain contributions proportional to cos 2 α, sin α cos α and sin 2 α. Those correspond to the coupling of the originalX boson with neutrinos and electrons, modified by its mixing with the SM Z-boson. In the limit of no mixing ( = δ = 0), they would be given by g 2 X Q L ν Q R /4 and g 2 X Q L ν Q L /4, respectively. Notice that the relevant parameters a 1 , a 2 , b 1 , b 2 have terms proportional to 2 g 2 , g X , g 2 X , g 2 , g X g etc. They have sometimes opposite signs, inducing interference effects which might be relevant depending on the values adopted for the parameters in a model [68]. The expressions above are exact and general. Thus the reader can easily reproduce our results and cast limits on any U (1) X model of interest.

Data Fitting
In this section, we use neutrino-electron scattering data to constrain U (1) X models and discuss the relevance of the Z mass to our reasoning. It turns out that the best limits can be obtained from reactor experiments TEXONO and GEMMA, and from the high energy beam experiment CHARM-II 5 Heavy Z : for Z s masses larger than 1.5 GeV, the effect of the Z is negligible for all considered neutrino-electron scattering experiments. Thus, the constraints can be described in terms of dimension-6 Fermi interactions. That said, the strongest constraint in this case should come from the experiment with the best measurement of electroweak parameters (e.g. sin 2 θ W ), which happens to be the CHARM-II experiment [78,79].
Light Z : For Z s lighter than 400 keV, the energy threshold of the detector dictates its sensitivity. The GEMMA experiment has a very low threshold for measuring the electron recoil energy and for this reason is expected to impose the best constraints on very light gauge bosons [80,81].
Intermediate Mass Z : For gauge bosons masses between 400 keV and 1.5 GeV the interplay between precision and energy threshold takes place, encompassing experiments such as TEXONO, LSND, and Borexino, etc. In this paper, we select three sets of data: GEMMA, TEXONO and CHARM-II, which should be quite representative of all neutrino-electron scattering data at various energy scales. Actually a previous study [68] shows that for the U (1) B−L model, the strongest constraint on the gauge coupling mainly comes indeed from these three experiments. Including other experiments such as LSND and Borexino has little effect on the combined constraint.
The details of the data relevant to our analyses are given next.
The CHARM-II experiment [78,79] used a horn focused ν µ (and ν µ ) beam produced by the Super Proton Synchrotron (SPS) at CERN. The mean neutrino energy E ν is 23.7 GeV for ν µ and 19.1 GeV for ν µ . From 1987 to 1991, 2677±82 ν µ e − and 2752±88 ν µ e − scattering events were detected, producing a very accurate measurement of the Weinberg angle sin 2 θ W = 0.2324 ± 0.0083. We take the data from [78] which published the measurement of the differential cross sections (in its Tab. 2). We thus directly use the cross section data to evaluate the χ 2 -values where S i /S i0 are the theoretical/measured differential cross sections, and ∆S i is the uncertainty. Neutrino and antineutrino data are combined together in the data fitting.  • TEXONO The TEXONO experiment [82] measured the ν e e − cross section with a CsI(Tl) scintillating crystal detector setting near the Kuo-Sheng Nuclear Power Reactor. Therefore the neutrino flux is the standard reactor ν e flux which peaks around 1 MeV. However, in TEXONO events are selected in the range 3 MeV < T < 8 MeV so the low energy part (E ν < 3 MeV) in the flux does not contribute to the signal. After data collection from 2003 to 2008, 414 ± 80 ± 61 events were selected. The measured Weinberg angle is sin 2 θ W = 0.251 ± 0.031(stat) ± 0.024(sys), and the ratio of experimental to SM cross section is 1.08 ± 0.21(stat) ± 0.16(sys). We perform a χ 2 -fit on the measured event rate R, where R i and R 0 i are the theoretical and measured even rates in the i-th recoil energy bin, and ∆R i is the corresponding uncertainty. Both R 0 i and ∆R i can be read off from Fig. 16 of [82]. The theoretical event rate is proportional to the event numbers divided by the bin width. The event number in the recoil energy bin T 1 < T < T 2 is computed by, Here a is an overall factor which can be calibrated using the SM values in Fig. 16 of [82], Φ(E ν ) is the reactor neutrino flux, and σ is a partial cross section defined as The integral can be analytically computed and the explicit expression is given in appendix C, which is technically useful in the data fitting. Note that for a neutrino of energy E ν , the recoil energy can not exceed  • GEMMA The GEMMA experiment [80,81] aimed at measuring the neutrino magnetic moment by a HPGe detector setting near the Kalinin Nuclear Power Plant. To reduce the SM background, only very low recoil energy events are selected, from 3 keV to 25 keV. In this range, the SM neutrino interactions are negligibly small with respect to its current sensitivity. We take the data from Fig. 8 of [81] and compute the event numbers also according to Eq. (4.3), except that the factor a is directly computed from the electron density in Ge, and the flux is renormalized to 2.7 × 10 13 events/cm 2 /s. The χ 2 -fit is the same as the TEXONO experiment.

Bounds
Now that we have described the data sets used in the analysis and the theoretical framework of U (1) X models, we can perform a χ 2 -fit to derive limits on the relevant parameters of the models, namely, (kinetic mixing parameter), g X (gauge coupling from the U (1) X symmetry), α (parameter that encodes the kinetic and mass mixing defined in Eq. (2.14), and m Z (gauge boson mass).
We will discuss now the constraints exhibited in Figs. 1-4. In Fig. 1 we show the limits on (m Z , g X ) in the absence of and α contributions, i.e. for = α = 0, which is equivalent to not having any kinetic and mass mixing terms. The results depend on explicit U (1) X models, i.e. the U (1) X charge assignments, which are listed in Tab. 1. Naturally, the larger the lepton charges under U (1) X the larger the new physics contribution to the neutrino-electron scattering, and thus the stronger the bound on the g X parameter. In the U (1) F model, the left-handed leptons have charge −3, whereas the right-handed electron has −4. Notice that due to these large charge assignments the U (1) F is subject to the strongest bound on g X . For m Z < 1 MeV, all models in our study are excluded for g X larger than 4 × 10 −6 . As for large gauge boson masses, say m Z = 10 GeV, g X > 6 × 10 −3 is excluded. The linear behavior of the limits for large Z masses as displayed in Fig.  1 occurs simply because m 2 Z 2m e T , see Eq. (3.14). Fig. 2 chooses a particular example, namely U (1) B−L , and studies the limits on (m Z , g X ) in the presence of nonzero α. We have fixed there the ratio of α and g X to certain values. If this ratio is around 1 the situation would correspond to the VEVs of the new scalar field lying close to the electroweak scale 6 . The plot shows that within −1 ≤ α/g X ≤ 1, the bounds become stronger when α increases by roughly a factor of 10 0.5 ≈ 3. We would like to comment that though α and g X are two independent parameters, they can be both very small at the same order of magnitude without fine tuning. As one can see from Eq. (2.15), in the limit of = 0, α is of the order δ/v 2 where v is the electroweak energy scale. If all the scalar VEVs are at (or not far from) the electroweak scale, then δ should be proportional to g X v 2 [see e.g. Eq. (A.12)], thus α is at the same order of magnitude as g X .
In Fig. 3 we present the limits on the kinetic mixing as a function of the Z mass for g X = α = 0. This is an approximate case. Notice from Eq. (2.15) that if , δ 1, then tan α ∼ s W , or α ∼ /2. 6 The case of ∼ g X seems less realistic and hence we do not show a corresponding plot different values of . for If α g X the limits are insensitive to α, for α g X we approach the situation displayed in Fig. 3.  Thus, the choice α = 0 as done in Fig. 3 is just an approximation. This setup is thus corresponding to a model with no mass mixing and the g X coupling finely tuned. Anyways, as before, it is quite visible the complementary role that GEMMA, TEXONO and CHARM-II play at probing light vector mediators.
In Fig. 4, we show an orthogonal scenario, namely bounds on the parameter α as a function of Z mass with g X , = 0. This case is relevant to a model featuring a sizable mass mixing, no kinetic mixing and a dwindled g X coupling. These bounds are independent of the U (1) X charges of the fermions and thus can be regarded being model-independent. We highlight that there are other relevant limits on the mass mixing parameter, such as those stemming from coherent neutrino-nucleus scattering or atomic parity violation, but they are not as restrictive as neutrino-electron scattering yet.

Conclusion
Additional neutral gauge bosons are a common feature of theories beyond the Standard Model. We have investigated here several different U (1) X models and have presented bounds on the key physical parameters (mass, gauge coupling and quantities describing mixing). The data we have used is from past experiments on neutrino-electron scattering, namely CHARM-II, GEMMA and TEXONO. We have provided general formulas for the Z-Z mixing and for the cross sections that allow to use them for any model with an additional Z boson or dark photon. Our study motivates analyses of upcoming neutrino-electron scattering data to further probe the parameter space of such models.

A Gauge Boson Mass Generation
In this appendix, we discuss the mass generation of the gauge bosons from the term where φ stands for all kinds of scalar fields with nonzero VEVs, denoted by φ .
According to the definition of D µ in Eq. (2.2), the mass terms of gauge bosons should be First, we consider the case that φ is an SU (2) L doublet. The hypercharge Q φ Y should be 1 or −1 to make possible, which is necessary to avoid a broken U (1) em . For Q φ Y = −1, we can redefine φ →φ = iσ 2 φ * to flip the sign of the hypercharge. So we only need to consider Q φ Y = 1. The VEV that does not break U (1) em should be where a complex phase ξ is allowed (e.g. in a CP violating 2 Higgs doublet model). The gauge boson mass matrix (in the fundamental basis) computed from Eqs. (A.3) and (A.1) turns out to bê Next, we consider a singlet φ. Similar to the argument around Eq. (A.2), we get Q φ Y = 0. The most general VEV is which leads to the mass matrixM 2 = diag 0, 0, 0, 0, Therefore, for arbitrary numbers of doublet and singlet scalar fields, the mass matrix should bê (A.10) The matrix in Eq. (A.7) can be block-diagonalized as follows, (A.14) The expression forM 2 in Eq. (A.11) is the most general mass term including mass mixing in scenarios in which U (1) em survives the symmetry breaking. For completeness, we finally give the exact expressions for the physical Z and Z boson masses, which read

B Cross Sections of Neutrino-Electron Scattering
Here we present the analytical calculation of the cross sections of (anti)neutrino-electron scattering in a general U (1) X model. The relevant Feynman diagrams are presented in Fig. 5  The initial/final states (momenta and spins) of neutrinos and electrons are denoted in the way shown in Fig. 5. The scattering amplitudes are written as follows: In the second lines of iM a1 and iM b1 , we have applied Fierz transformation to get uniform expressions so that they can be combined with the NC contributions.
The total amplitudes of ν e + e − scattering and ν e + e − scattering are Note that in realistic experiments the incoming (anti)neutrinos should be (right-)left-handed. So in practice, one can attach the right-handed or left-handed projectors P R = 1+γ 5 2 , P L = 1−γ 5 2 to the initial state of incoming antineutrinos or neutrinos respectively: Applying the trace technology, we get and c i , d i ,c i , andd i are defined by One can compute |iM b | 2 in the similar way. The result is , (B.29) A − ≡ −c W g X Q L − g s W c W g X Q L ν + g s W 2 ( 2 − 1) , (B.30) So far we have not taken any approximation in the above calculation. Since most neutrinoelectron scattering data are at energies much lower than m W and m Z , we will take the approximation If the contribution of diagram (a1) or (b1) in Fig. 5 is absent, one simply applies the limit In the approximation given by Eq. (B.32) and Eq. (B.33), G ± can be expressed (we also assume Q L ν = Q L ) as , (B.34) where G SM + and G SM − in the limit α → 0 are pure SM contributions: (B.36) Note that A ± and B ± are suppressed by and g X :

C Partial Cross Section
The partial cross section is defined as We can rewrite it as where I is a matrix defined as the integral of the matrix K. The matrix K and the vector x are given as follow: x ≡ (g SM 1 + a 1 , g SM 2 + a 2 , b 1 , b 2 ) (for neutrino) (g SM 2 + a 2 , g SM 1 + a 1 , b 2 , b 1 ) (for antineutrino) , (C.3) (C.4) The nonzero analytical expressions of I ij ≡´T 2 T 1 K ij dT are: (C.5) , (C.11) . (C.13)