Atomic Parity Violation, Polarized Electron Scattering and Neutrino-Nucleus Coherent Scattering

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Nuclear Physics B 959 (2020) 115158 www.elsevier.com/locate/nuclphysb

New physics probes: Atomic parity violation, polarized electron scattering and neutrino-nucleus coherent scattering

Giorgio Arcadi a,b, Manfred Lindner c, Jessica Martins d, Farinaldo S. Queiroz e,∗

a Dipartimento di Matematica e Fisica, Università di Roma 3, Via della Vasca Navale 84, 00146, Roma, Italy b INFN Sezione Roma 3, Italy c Max-Planck-Institut für Kernphysik (MPIK), Saupfercheckweg 1, 69117 Heidelberg, Germany d Instituto de Física Teórica, Universidade Estadual Paulista, São Paulo, Brazil e International Institute of Physics, Universidade Federal do Rio Grande do Norte, Campus Universitário, Lagoa Nova, Natal-RN 59078-970, Brazil Received 7 April 2020; received in revised form 22 July 2020; accepted 20 August 2020 Available online 27 August 2020 Editor: Hong-Jian He

Abstract

Atomic Parity Violation (APV) is usually quantified in terms of the weak nuclear charge QW of a nucleus, which depends on the coupling strength between the atomic electrons and quarks. In this work, we review the importance of APV to probing new physics using effective field theory. Furthermore, we correlate our findings with the results from neutrino-nucleus coherent scattering. We revisit signs of parity violation in polarized electron scattering and show how precise measurements on the Weinberg’s angle give rise to competitive bounds on light mediators over a wide range of masses and interactions strengths. Our bounds are firstly derived in the context of simplified setups and then applied to several concrete models, namely Dark Z, Two Higgs Doublet Model-U(1)X and 3-3-1, considering both light and heavy mediator regimes. © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

* Corresponding author. E-mail addresses: [email protected] (G. Arcadi), [email protected] (M. Lindner), [email protected] (J. Martins), [email protected] (F.S. Queiroz). https://doi.org/10.1016/j.nuclphysb.2020.115158 0550-3213/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158

1. Introduction

For decades it was assumed that the laws of nature preserved parity, but the seminal paper of Lee and Yang in 1956 gave rise to a different perspective [1]. It was indeed confirmed in 1957 in the realm of weak interactions, via the beta decay in Cobalt [2] and muon decay [3]. In 1959 the possibility of observing parity violation in atomic physics and electron scattering was contemplated [4] and further investigated in the late 70’s [5–7]. Interestingly these ideas preceded the theory of electroweak interactions. The following decades were populated by experiments that aimed at probing parity violation [8]. For interesting reviews on APV see [9–14]. Our current understanding of parity and APV has greatly improved. Given the experimental precision acquired over the years, we may now test new physics models that feature parity- violating interactions. The main objects of study in this regard are the Weak Charge of the Cesium (Cs) and polarized electron scattering. Concerning the first observable, the weak charge of a nucleus, QW , is an analogous of the electromagnetic charge, where the Z boson is the key player of the atomic electron and nucleus interactions instead. QW is the sum of the weak charges of all constituents of the atomic nucleus, QW = (2Z + N)QW (u) + (Z + 2N)QW (d), where QW (u, d) accounts for the Z interactions 2 with up and down quarks and it depends on sin θW , with θW being Weinberg’s angle. To un- derstand how the Z boson can affect atomic transitions and QW is extracted from experiments, one needs to perform precise atomic physics calculations and measure the left-right asymmetry 2 2 2 ∼ −15 ALR, which is naively estimated to be ALR α me/mZ 10 . Despite atoms with a high atomic number be, in principle, better suited for experimental observation of APV [15], because of the enhancement in ALR by orders of magnitude, such high atomic number makes the theoretical determination of ALR extremely challenging. For this reason, Cesium has become a popular target because it offers a good compromise between high atomic number, necessary to have siz- able effects, and relatively simple atomic structure, required to make precise atomic calculations. Precise measurements of APV in Cesium have been presented in [16,17]. Based on them, de- exp =− ± tailed atomic physic computations yielded the following determination: QW 72.62 0.43 th =− ± [18,19], in slight disagreement with the SM prediction QW 73.23 0.01 [20,21]. Such precise measurements of the Cs weak charge can be used to constraint new physics effects. The other parity violation observable, polarized electron scattering, also constitutes an important laboratory to new physics searches. Again, the left-right asymmetry is the key observable. For deep inelastic scattering processes of the type eL,RN → eX, the left-right asymmetry can be expressed, in the quark model and in the limit of zero nucleon mass, in the relatively simple form 2 1 A/Q = a1 + a2f(y). The coefficients a1, a2 depend on the axial-vector coupling between the 2 electron and quarks, which, in turns, depends on sin θW , θW being Weinberg’s angle. y and Q represent, respectively, the fraction of energy transferred from the electron to the hadrons and the momentum transfer. Detailed expressions for a1, a2 and f can be found, for example, in [8,22]. 2 Measurement of A translates into a measurement of sin θW at a given momentum Q. It is well- known that photon exchange diagrams conserve parity but processes mediated by the Z do not as it not interact with left-handed and right-handed fermions in the same way. In a similar vein, eventual additional massive vector bosons from new physics models might also contribute to the left-right asymmetry. This kind of contribution can be conveniently parametrized as a shift on

1 In the case of electron-positron scattering or the so-called Moeller scattering (scattering on electrons), the Q2 dependence of the asymmetry is more complicated and is parametrized by a form factor. G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 3

2 Fig. 1. Scale dependence (gray curve) of sin θW [27,28] compared with measurements (colored points) from APV [29] as well as the E158 [30], Qweak [31,32], P2 [33], Mesa [34], Moller [35], Solid [36]. (For interpretation of the colors in the ﬁgure(s), the reader is referred to the web version of this article.)

2 sin θW (see e.g. [23–26]for previous analyses along this line). If measurements of A translate 2 into a determination of sin θW compatible with the Standard Model (SM) prediction, one can obtain constraints on New Physics effects responsible for parity violation possibly contributing to the left-right asymmetry. 2 We show in Fig. 1 an illustration of the expected energy scale dependence of sin θW (see e.g. 2 [27,28]for details) together with different measurements of sin θW (see also [21]). The aforementioned observables depend on the couplings of the New Physics degrees of freedom with electrons and quarks. Constraints on parity violation effects will be then translated into limits on such couplings. That has been the whole story up to now, but with the observation of neutrino-nucleus coherent scattering new information came into light. Strictly speaking, neutrino-nucleus coherent scattering and parity violation probes are sensitive to different interactions, between electron and quarks in the former case, between neutrinos and quarks in the latter. However, due to SU(2) invariance, it is reasonable to assume that new degrees of freedom can couple with both electrons and neutrinos (one notable exception is represented by the so-called dark photon though). In this work, we will revisit this scenario and explore the possible complementarity between parity violation observables and coherent neutrino scattering [23–26]. To be as general as possible, the first part of our analysis will be carried out adopting an EFT approach as well as a simplified model in which the SM spectrum is extended by a new light spin– 1 mediator. We will apply later on our findings to some more complete models already existing in the literature. While this approach is not new by itself, we stress again that our work makes an advance concerning the existing literature in light of the considered complementarity between neutrino-nucleus coherent scattering, APV, and polarized electron scattering. At the same time, it is worth pointing out that the models investigated here might be embedded in non-trivial neutrino sectors, e.g. with the presence of sterile neutrinos, which can weaken the correlations between neutrino coherent scattering and parity violation phenomena. We will not consider this possibility here. The paper is structured as follows. In Section 2, we will review the theoretical aspects of parity violation; in Section 3 we discuss APV; in Section 4 we address the complementary aspects with neutrino-nucleus coherent scattering using effective field theory; in Section 5 we study polarized electron scattering in terms of light mediators and put into perspective with neutrino-nucleus 4 G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 coherent scattering. Lastly in Section 6 we discuss our bounds using concrete models proposed in the literature.

2. Parity violation

Parity violation effects can be described through effective lagrangian [20,37], 2 + 2 PV g g 1 μ 5 1 2 2 −L = eγ¯ γ e − sin θW uγ¯ μu eff m2 4 4 3 Z 1 1 + − + sin2 θ dγ¯ d + 4 3 W μ 1 + eγ¯ μγ 5e f uγ¯ u + f dγ¯ d , (1) 2 Vu μ Vd μ with the first two lines describing the SM contribution and the New Physics effects represented by analogous operators, as the SM ones, but containing three new parameters: the energy scale and the two adimensional couplings fVu and fVd. While we could have reduced the number of free parameters by absorbing the coupling constants into the New Physics scale, this parametrization eases the application of our findings to more concrete models. Now that we have reviewed how one can describe parity violation using effective field theory we will link it to APV.

3. Atomic parity violation

As we want to relate parity violation to APV, we need to define the weak charge of a nucleus eff which enters into the Hamiltonian of the electron field [37]. This weak charge, QW is the sum of both Standard Model and New Physics contributions. The former is given by: SM = − 2 − QW (Z(1 4sin θW ) N), (2) up to radiative corrections. In analogy, we can define a New Physics contribution by combining the couplings of the vectorial currents: f f f f QNP = Z 2 Vu + Vd + N Vu + 2 Vd W 2 2 2 2 3 = f eff(Z + N), (3) 2 Vq eff where the effective coupling fVq is defined as: f (2Z + N) + f (Z + 2N) f eff = Vu Vd (4) Vq 3(Z + N) We can then write an effective Hamiltonian in terms of the new effective charge. In the non- relativistic regime, after a Fourier transform the propagator becomes: − || − || 1 e mZ r 1 m2 e mZ r → = Z , (5) 2 − 2 || 2 || mZ q 4π r mZ 4π r G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 5

2 → 2 →∞ when mZ we get, − || 1 m2 e mZ r δ(r) Z → . (6) 2 || 2 mZ 4π r Then, the effective Hamiltonian describing Atomic Parity Violation can be written in terms of the weak charge of the nucleus as: PV =−LPV Heff eff int int g2 + g 2 1 3 = e†γ 5e QSM + f eff(Z + N) δ(r) (7) 4m2 4 W 2 Vq Z g2 + g 2 1 16m2 1 = e†γ 5e QSM + Z QNP δ(r) (8) 2 W 2 + 2 2 W 4mZ 4 g g G =e†γ 5e √F Qeff(Z, N)δ(r). (9) 2 2 W We can deﬁne the variation of the effective charge, with respect to the SM expectation, as: √ = eff − SM = 2 2 3 eff + QW QW QW 2 fVq(Z N). (10) GF This should be compatible with experimental observation. As already mentioned, the best probe 133 of APV effects is represented, at the moment, by transitions on stable isotope 78 Cs. From this, we can determine the following bound:

0.18 ≤ QW ≤ 1.17 (11) −5 −2 Inserting GF = 1.1663787 × 10 GeV , Z = 55, N = 78, we constrain the ratio effective coupling over the energy scale: eff − − fVq − − 1.1 × 10 9 GeV 2 ≤ ≤ 1.85 × 10 9 GeV 2. (12) 2 From the relation above it would possible to determine a relation between the fVu and fVd parameters, as shown in Fig. 2. The EFT description illustrated above is valid as long as the new degrees of freedom associ- ated with the New Physics scale is larger than the typical energy/momentum transfer measured in parity violation. In the simplest case, ≡ mZ with Z a new spin-1 boson (we will brieﬂy illustrate some concrete models in section 6. If this is not the case, namely mZ 100 MeV [37], atomic parity violation effects are damped by a form factor K(mZ) < 1to account for the propagator of the new boson. This feature should be included in eq. (10). We will not explore this case here. An important complementary probe for light bosons rises from polarized electron scattering. Before discussing this, we will compare, in the EFT limit, the bounds from APV with the ones stemming from neutrino-nucleus coherent scattering.

4. Neutrino-nucleus coherent scattering

According the discussion the EFT approach discussed in Section 3, APV is described by μ four ﬁeld operators of the form eγ¯ μγ5eqγ¯ q, with q = u, d, hence connecting u, d quarks with 6 G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158

2 Fig. 2. Bound on the couplings between the electron neutrino and up-quark (fVu/ ) and electron neutrino and down 2 quark (fVd / ) as deﬁned in Eq. (13). The green area is the region allowed by the COHERENT data. The CHARM constraint stems from νe − N inelastic scattering data [55]. The LHC bound results from missing energy searches [56]. The very narrow band corresponds, ﬁnally, to the constraint from APV in Cesium. electrons. Following the generic SU(2) invariance argument mentioned in the introduction, we will assume that the New Physics originating APV is also responsible for interaction between quarks and neutrinos of the form: f f −L = Vu(ν¯ γ μν )(uγ¯ u) + Vd(ν¯ γ μν )(dγ¯ d), (13) eff 2 L L μ 2 L L μ which can provide a microscopic description of neutrino-nucleus scattering processes. Indeed the most general basis of dimension-6 operators coupling a lepton pair and a quark pair is given by [38–42]:

1 glq μ L = = (l γ l )(q γ q ) D 6 2 2 L L L μ L 1 g 1 g + eu (e γ μe )(u γ u ) + ed (e γ μe )(d γ d ) 2 2 R R R μ R 2 2 R R R μ R g g + lu (l u )(u l ) + ld (l d )(d l ) 2 L R R L 2 L R R L g g + qe (q e )(e q ) + qle (l e )(q u ) 2 L R R L 2 L R L R g gql + le (l e )[(q )T u ]+ (q e )[(l )T u ] (14) 2 L R L R 2 L R L R 2 with qL, lL being the SM SU(2) quark and lepton doublets and = iτ . From this both the NP operator responsible for APV and eq. (13) originate with: 1 1 f = (g − g − g + g ), f = (g − g − g + g ) Vu 8 eu qe lq lu Vd 8 ed qe lq ld 1 1 f = (g − g ), f = (g − g ) (15) Vu 4 lu lq Vd 4 lq ld In order to obtain the previous relations we have used the Fierz rearrangement:

μ (ARγ BR)(CLγμDL) =−2(CLBR)(ARDL) (16) G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 7

Moreover, it is worth noticing that for Majorana neutrinos, the current in eq. (13) has the same structure of the electron current in the parity violation operator in eq. (1). Concerning neutrino-nucleus scattering can be probed by the COHERENT experiment [43–45], for example. Given the agreement with the Standard Model prediction, we can use the COHERENT data to constrain new physics effects. In the EFT framework under consideration, this constraint can be expressed through the following relation [46,47]:

V + + + V + + Z gp 2 Vu Vd N gn Vu 2 Vd

=± V + V Zgp Ngn (17)

V where gp,n are the couplings of the SM Z-boson with the proton and the neutron: 1 1 gV = − 2sin2 θ ,gV =− (18) p 2 W n 2 while: √ √ f f = 2 Vu = 2 Vd Vu 2 , Vd 2 (19) GF GF This equation is solved for: A + N =− (20) Vu A + Z Vd and

V + V A + N 2 Zgp Ngn =− − , (21) Vu A + Z Vd A + Z where A = Z + N. Consequently, the allowed regions from the COHERENT data would ap- f f pear as linear band in the bidimensional plane Vd , Vu . Assuming, for simplicity, f = 2 2 Vu = fVu, fVd fVd, we can compare the sensitivity of the coherent experiments with APV. This kind of comparison is shown in Fig. 2. The green area is the allowed region in the fVu vs fVd plane [47]. We remark again that the solution illustrated above is valid as long as the New Physics scale is larger than the typical momentum transfer q in the neutrino scattering processes. If this is not the case, one should explicitly consider the new BSM mediator of the interactions between neutrinos and SM quarks and hence the following redefinition: √ √ 2 fVq 2 fVq → (22) G 2 G 2 + 2 F F mZ q where fVq should be now interpreted as the product of the couplings of the mediator with neutrinos and up/down quarks. Having in mind the importance of neutrino-nucleus interactions as a probe for New Physics, several studies have been carried out [48–53]. In this work, we want to compare the sensitivity of neutrino-nucleus coherent scattering and APV. To this purpose we need to compute f eff , Vq as defined in eq. (4) with N = 78 and Z = 55. The APV observable is related to the coupling between electrons and quarks. Assuming, for simplicity, that the couplings in eq. (13)are equal 8 G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 to the ones responsible for APV, we can pick the pairs fVu, fVd inside the overlapping region in Fig. 2. The green band delimits the region allowed by the COHERENT data [54], whereas the blue one comes from the CHARM experiment based on the observation of νe − N inelastic scattering [55]. The effective four field interactions responsible for neutrino coherent scattering may be also tested by LHC searches of monojet events [56]. Under our assumption, the latter can then provide a further complementary bound (shown as a gray region in Fig. 2) to the ones from parity violation processes. Collectively these different data sets restrict new physics to live inside the overlapping region, which roughly implies that |f eff /2| < 10−6. Therefore, if couplings involved in APV and Vq neutrino-nucleus coherent scattering processes for some reason are similar, APV clearly constitutes a more promising probe. Be that as it may, we emphasize that the couplings involved in these processes can be quite different depending on the model. Anyway, it is exciting to see that COHERENT, a 14 kg detector, can already place important bounds on new physics. This fact has triggered several new physics sensitivity studies using COHERENT data and other nuclei [50,53,54,57–65]. There are upcoming experiments that aim at probing neutrino-nucleus coherent scattering at different energies, which will certainly yield complementary tests to new physics [66–68]. We have parametrized the New Physics effect in terms of the effective couplings and the energy scale . This parametrization is valid in the regime in which the new physics scale is much heavier than the typical energy scale involved in the measurement. However, New Physics can also appear as light mediators, here generically dubbed as (Z), namely with masses much below one of the SM Z-boson. This kind of scenario can be effectively probed through polarized electron scattering, as illustrated in the next section.

5. Polarized electron scattering

Atomic Parity Violation constitutes an important probe for New Physics. If the latter, however, surfaces at low energy, polarized electron scattering becomes an ideal laboratory. Indeed, low- energy scattering of polarized electrons on electrons and other targets are very sensitive to parity violation effects at low energy and, consequently, to the presence of light mediators. It is useful, in particular, to test scenarios in which a new spin–1 degree of freedom, possibly the gauge boson of a new U(1) symmetry, features kinetic and mass mixing with the SM Z-boson. As already pointed, we will adopt, ﬁrst, a generic description of the interactions of the Z boson in terms of a kinetic mixing parameter , deﬁned by: L ∈ μν B Zμν (23) 2 cos θW and a generic mass mixing parameter δ: m2 −δm m M2 = Z Z Z , (24) − 2 δmZmZ mZ where 0 ≤ δ<1. If the Z is light compared to Z the mass matrix can be rewritten as: 1 − Z 2 2 2 M = m m , (25) − Z Z Z 2 mZ with G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 9

mZ Z = δ . (26) mZ The cross-term (23) can be rotated away leading to the following redefinition of the Z and photon fields: → + Aμ Aμ Zμ → − Zμ Zμ tan θW Zμ (27) inducing, in turn, the following interactions for the Z boson: L =− em − g NC μ e Jμ ZJμ Z , (28) 2 cos θW NC where Jμ is the Standard Model neutral current. These terms induce weak currents that can accounted for by redefining sinθW as [24,26,69],

2 → 2 = − 2 cos θW 2 2 sin θW κd sin θW ,κd 1 δ f(Q /mZ ), (29) Z sin θW 2 2 where f(Q /mZ ) is the propagator effect given by [70–73],

2 2 1 f(Q /m ) = . (30) Z + 2 2 1 Q /mZ Using equations (29) and (30)we can write the change of the weak angle due to the mixing between Z and Z as, 2 − mZ 2 2 sin θW 0.42 δ f(Q /mZ ), (31) mZ and so we can put bounds on the mixing given the difference between measurements and prediction of the weak angle, 2 2 5.67 2 2 mZ 2 2 2 ( sin θ ) (1 + Q /m ) δ2 W m Z Z 2 2 2 m + Q 5.67 2 2 Z 2 ( sin θW ) . (32) δ mZmZ

2 A summary of bounds from measurements of sin (θW ) is provided in the Table 1. We highlight this framework is inspired by the works done in [23–26]. From Table 1 we notice that the bounds on become stronger for large values of δ which accounts for the mass mixing. We exhibited these bounds for several values of δ in Fig. 3. Since the experiments run at different energies they are sensitive to different Z masses. In particular, SoLID is very sensitive to Z masses around 2 1GeV. It is remarkable the precision aimed by Moller at JLab planning to measure sin (θW ) to ±0.00029 at Q = 75 MeV, followed by the P2 experiment with precision of ±0.00033 in 2 sin (θW ) for Q = 67 MeV. Looking either at the Table 1 or Fig. 3 one can see that if the mass −2 2 mixing parameter δ is of the order of 10 precise measurements on sin θW give rise to stringent bounds on , namely 2 < 10−4. 2 2 Notice that in the regime mZ Q bounds rising from atomic parity violation are also applicable. These can be straightforwardly accounted for by the following rescaling in the parameters of the weak interaction lagrangian: 10 G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158

Table 1 90% conﬁdence level bounds on the kinetic mixing parameter for light mediators for different experiments 2 that aim at measuring sin θW at low energies. All masses are in MeV units. 2 Lab Q sin θW (mZ) Light Mediator (90% CL) 2 2 2 × −5 m +160 E158 160 MeV 0.2329(13) 2 < 1.54 10 Z δ2 mZmZ 2 2 2 × −6 m +170 Qweak 170 MeV ± 0.0007 2 < 2.78 10 Z δ2 mZmZ 2 2 2 × −7 m +75 Moller 75 MeV ± 0.00029 2 < 4.77 10 Z δ2 mZmZ 2 2 2 × −7 m +67 P2 67 MeV ± 0.00033 2 < 6.17 10 Z δ2 mZmZ 2 2 2 × −6 m +2500 SoLID 2.5 GeV ± 0.0006 2 < 2.04 10 Z δ2 mZmZ

GF → ρd GF 2 2 sin θW → κd sin θW (33) where: 2 m ρ = 1 + δ2 Z d 2 + 2 mZ Q 2 1 m κ = 1 − δ2 Z (34) d θ 2 + 2 Z tan W mZ Q 2 SM The shifts in the GF and sin θW parameters imply a modiﬁcation of QW : SM →− + 2 + 2 QW 73.16(1 δ ) 220 δ cos θW sin θW (35) Z Agreement, within 1σ , with measurements in Cesium hence requires:

|δ2 1 − 1.27 | 0.005 (36) Z Similarly to what done for atomic parity violation, we would like to compare the limits stem- 2 ming from sin θW to those from neutrino-nucleus coherent scattering. This task is not trivial though, since the relevant parameter, namely , Z, δ are in principle independent. We will then compare bounds from polarized electron scattering and neutrino-nucleus coherent scattering for some speciﬁc models, illustrated in the next section, where it is possible to establish relations between these parameters.

6. Models

In this section, we will interpret the bounds from parity and atomic parity violation, expressed, until now, in terms of generic parameters, within a collection of anomaly free models proposed in the literature. G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 11

Fig. 3. Bounds of the experiments E158, Qweak, Moller, P2 and SoLID on the mixing between Z and Z with respect to the mass of the new light neutral gauge boson following the relations on the table. In each plot the mixing parameter δ is − changing logarithmically from δ = 10 4 to δ = 1, so we can also visualize the dependence of with this parameter.

6.1. Dark hypercharge and dark Z models

One of the simplest models which can be probed by parity violation phenomena and neutrino elastic scattering is the one proposed in [24,74,75]. The Z is the gauge boson of a new abelian symmetry. The kinetic mixing operator eq. (23)is generically present in such scenario μν since the operator BμνX is gauge invariant. The eventual mass mixing parameter δ depends, instead, on the details of the sector, for example, a dark Higgs sector, responsible for the Z 12 G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 mass. In the absence of specifications of this sector, the parameter δ can be regarded as free. Under the assumption that the SM fermions are not charged under the new abelian symmetry, the phenomenology of the new boson is fully specified by its mass and the value of the and δ parameters. Under these assumptions, the considered scenario, dubbed dark hypercharge model in [58], substantially coincides with the simplified model presented in the previous subsection. Fo- cusing, for simplicity, on the case of a light mediator, one can apply the constraints summarized 2 −7 −2 in Table 1. We see, for example, that we get < 10 for mZ ∼ 100 MeV and δ ∼ 10 , using the P2 projected sensitivity. This limit is slightly stronger than the one achieved using BaBar data [76] which is the relevant experiment at this mass range. Therefore, we conclude that our bounds are stronger. Ref. [58] considered a variant of the model, consisting of the limit → 0, dubbed dark Z since, in this case, the couplings of the Z are proportional to the ones of the SM gauge boson. 2 In such a case, we have no shift in sin θW , but bounds from APV and COHERENT are still effective. As shown in [58], APV gives the strongest constraint in the 0.5 mZ 5 GeV range.

6.2. Two Higgs doublet model augmented with Abelian symmetry

An interesting UV completion of the scenario presented before consists into a Two Higgs doublet model extended with a gauged U(1)X symmetry [24,77,78]. In this class of models the mass of the Z and the δ parameter are not free but are determined by the new gauge coupling gX, the vevs v1,2 of the two Higgs doublets and the vev vX of a Higgs singlet responsible of the spontaneous breaking of the U(1)X. More speciﬁcally we have that:

m = g v cos β2/δ (37) Z X = = 2 + 2 where tan β v2/v1, v v1 v2/2 and:

2 cos β cos βX δ = (38) 2 + 2 2 2 − 2 − 2 qX cos βX sin β QX1 QX2 qX with QX1, QX2 being the (eventual) charges of the two doublets under U(1)X. The parameter remains, in general, free, since the kinetic mixing operator is allowed, by gauge invariance, at tree level. The 2HDM+U(1)X has been extensively analyzed in [78] considering both the light and heavy mediator regimes. In the first case one typically finds, from most of the assignments of the U(1) charges, δ ∼ 10−2 assuming tan β ∼ 50. Using this value for δ, we find again the bound 2 −7 < 10 for mZ 100 MeV, illustrated in the previous subsection. In the heavy mediator ∼ 2 2 ≤ regime, we can apply our effective field theory approach taking fVu fVd getting gX/mZ −9 4 4.38 × 10 GeV or, equivalently, mZ ≥ 1.5gX × 10 GeV. This bound is applicable under the assumption that axial-vector couplings between the electrons are present as occurs for many models discussed in [78]. Having in mind that LEP bound on vector mediators roughly reads, 3 mZ > 7gX × 10 GeV [79], we conclude that APV provides a stronger bound. This limit from LEP was derived for the B-L model where only vectorial interactions are present but it is roughly applicable to other models [80,81]. Anyway, our conclusion stands, APV gives rise to a more restrictive bound on the Z mass. One may wonder about the LHC lower mass bounds on such vector bosons. It has been shown that many of these models predict a large Z width. This feature weakens LHC sensitivity. Analyzing LHC data it has been found that mZ > 1 − 2TeV for many models taking gX = 0.1[82], which is again weaker than APV. In summary, APV seems to be the most promising laboratory for such models as far as the Z mass is concerned. G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 13

Table 2 Summary of the bounds we found applied to concrete models. See text for the details concerning the assumption behind these limits. Model Bound 2 −7 ∼ Dark Hypercharge < 10 ,formZ 100 MeV −4 ∼ Dark Z Z < 10 ,formZ 0.5GeV 2 −7 ∼ 2HDM-Light Z < 10 ,formZ 100 MeV × 4 2HDM-Heavy Z mZ > 1.5 gX 10 GeV, for mZ 1GeV 3-3-1 Model mZ > 1.7TeV 2 −10 ∼ Light Z Models < 10 ,formZ 100 MeV

An interesting variant of the 2HDM+gauged U(1) has been recently proposed in [83], suc- cessfully accommodating neutrino masses within the type II seesaw. The mass of the Z comes from the vacuum expectation value of the scalar doublets and therefore the Z must be light. According to our reasoning, the best probe, in the context of parity violation processes, is represented by polarized electron scattering. Given the charge assignments reported in Table 1 of −1 2 −10 [83], we get δ ∼ 10 . For such a large value of δ we ﬁnd < 10 for mZ 100 MeV using 2 −7 the P2 experiment, and < 10 for mZ 1GeV using SoLID projection. These bounds are much stronger than those derived using The Heavy Photon (HPS) Search Experiment and Belle projections shown in [83].

6.3. 3-3-1 model

Another class of realistic frameworks embedding new Z bosons is represented by the 3-3-1 models (Table 2). They are based on the SU(3)c ⊗ SU(3)L ⊗ U(1)N gauge group [84,85] and can be used to explain the number of replication of fermion generations in the Standard Model and are, as well, able to address neutrino masses and dark matter. The presence of the U(1)N group gives rise to heavy Z whose mass is set by the energy scale at which the 3-3-1 symmetry is broken down to the Standard Model gauge group. The Z does have axial-vector couplings to electrons and therefore might leave imprints on APV. Although, the Z couplings to fermions are suppressed, of the order of 10−2. Bounds from parity violation processes can be applied to 3-3-1 models, in the EFT regime. We can then compute f eff using the expressions for the Vq vector and axial-vector couplings of the Z as given e.g. in Table V of [86]. Ref. [86] considers different realizations of the 3-3-1 model distinguished by the value of the parameter√ β in the charge operator Q = T3 + βT8 + N. For the dubbed model A, identified by β = 3, we find a the lower mass bound mZ > 1.7TeV. This limit is, however, sub-dominant when compared to existing bounds stemming from dijet and dilepton searches at the LHC which impose mZ > 4TeV [87–90]. Interesting bounds are set, as well, by other observables like flavor physics but they are not as relevant [91–96]. We highlight that there are possible extensions of this model via the inclusion of right-handed neutrinos which can weaken the LHC bounds by decreasing the Z branching ratio into charged leptons and quarks [97]. In that case, our bounds could become competitive.

6.4. Lμ − Lτ models

Models based on the Lμ − Lτ gauge symmetry have recently brought a lot of attention due to some ﬂavor anomaly [98–104]. The Z boson can be quite light and has no interactions with 14 G. Arcadi et al. / Nuclear Physics B 959 (2020) 115158 quarks at tree-level. At loop-level, one could nevertheless generate the neutrino-nucleus coherent scattering and the parity-violating observables discussed here. Albeit, there are already stringent bounds rising from neutrino-trident production and meson mixings [105,106], making our as- sessment of 1-loop induced parity violation effects not relevant, in agreement with [58].

7. Conclusions

We have reviewed the theoretical aspect of parity violation and put it in context with other relevant observables. We treated Atomic Parity Violation using effective ﬁeld theory and showed how one can constrain New Physics via precise measurements of the Cesium weak charge. Moreover, we have discussed neutrino-nucleus coherent scattering and shown that Atomic Parity Viola- tion leads to a more restrictive bound on the new physics scale under the assumption that the new physics particle couples to electrons and neutrinos with similar strength. This conclusion is also valid for heavy vector mediators with masses at the TeV scale, for instance. Shifting the discussion to light mediators, we have parametrized new physics effects in polarized electron scatterings in terms of the sin θW and explored the sensitivity of new measurements on sinθW to derive bounds on the kinetic mixing between the Z and Z gauge bosons as a function of the Z mass. Lastly, we applied our ﬁndings to existing models in the literature.

CRediT authorship contribution statement

FSQ, ML and GA were responsible for the conceptualization. JM and GA carried out the formal analysis. All authors contributed to the writing.

Declaration of competing interest

The authors declare that they have no known competing ﬁnancial interests or personal rela- tionships that could have appeared to inﬂuence the work reported in this paper.

Acknowledgements

The authors thank Omar Miranda, Alex Dias, Carlos Pires, Diego Cogollo, Paulo Ro- drigues, and Clarissa Siqueira for feedback on the manuscript. FSQ acknowledges support from CNPq grants 303817/2018-6 and 421952/2018-0, UFRN, MEC and ICTP-SAIFR FAPESP grant 2016/01343-7. JM thanks the support from CAPES grant. We thank the High Performance Com- puting Center (NPAD) at UFRN for providing computational resources.

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