First-Generation New Physics in Simplified Models: From Low-Energy Parity Violation to the LHC

New-physics (NP) constraints on first-generation quark-lepton interactions are particularly interesting given the large number of complementary processes and observables that have been measured. Recently, first hints for such NP effects have been observed as an apparent deficit in first-row CKM unitarity, known as the Cabibbo angle anomaly, and the CMS excess in $q\bar q\to e^+e^-$. Since the same NP would inevitably enter in searches for low-energy parity violation, such as atomic parity violation, parity-violating electron scattering, and coherent neutrino-nucleus scattering, as well as electroweak precision observables, a combined analysis is required to assess the viability of potential NP interpretations. In this article we investigate the interplay between LHC searches, the Cabibbo angle anomaly, electroweak precision observables, and low-energy parity violation by studying all simplified models that give rise to tree-level effects related to interactions between first-generation quarks and leptons. Matching these models onto Standard Model effective field theory, we derive master formulae in terms of the respective Wilson coefficients, perform a complete phenomenological analysis of all available constraints, point out how parity violation can in the future be used to disentangle different NP scenarios, and project the constraints achievable with forthcoming experiments.


Introduction
The Standard Model (SM) of particle physics has been very successfully tested and confirmed with great precision in the last decades with the Higgs discovery in 2012 unveiling the last missing piece. While the LHC has not (yet) found any new particles directly, precision experiments are becoming increasingly important to gather hints as to how the SM needs to be extended, so as to eventually construct a more fundamental theory that can account for dark matter, neutrino masses, and new sources of CP violation. Such measurements can usually be carried out to the highest precision when particles composed of first-generation quarks and leptons are involved, given the practical life-time constraints of other SM sectors.
One class of low-energy precision experiments concerns parity violation (PV), as here QCD and QED processes that otherwise overshadow weak SM and potential NP effects are suppressed. PV is realized in atomic parity violation (APV), parity-violating electron scattering (PVES), and, more recently, coherent neutrino-nucleus scattering (CEνNS). Next, β decays are sensitive to modifications of the charged current, with most precise constraints available from superallowed β decays and neutron decay. Further constraints arise from electroweak precision observables (EWPO) at the Z pole, and finally also at the LHC precision-frontier measurements such as non-resonant di-lepton searches are possible. This broad class of complementary measurements motivates combined analyses, especially once hints for NP arise in one or more of the processes, to elucidate which, if any, NP interpretations are viable.
While the anomalies in semi-leptonic B decays and the anomalous magnetic moment of the muon point towards NP related to second-and third-generation fermions, the CAA and the CMS di-lepton excess can be related to first-generation quarks and leptons, with simultaneous explanations possible in terms of the effective dimension-6 operator Q (3) q [62]. Similarly, explanations of the CAA via modified W -u-d couplings also require NP related to first generation-quarks [1,[63][64][65]. In this paper, we take the large array of complementary measurements sensitive to first-generation NP, together with hints for potential NP effects, as motivation to perform a combined analysis, concentrating on possible correlations among the processes listed above.
In general, however, most of these processes cannot be correlated in a model-independent way, 1 due to the proliferation of independent Wilson coefficients in SM effective field theory (SMEFT) [69,70]. For this reason, we will consider a set of simplified NP models, covering the four different classes of new particles [71] that can give rise to modified vector (or axial-vector) quark-lepton interactions below the EW symmetry breaking scale: leptoquarks (LQs), vector bosons (VBs), vector-like quarks (VLQs), and vector-like leptons (VLLs). In particular, we do not consider particles giving rise to scalar interactions (in the qq basis), as the resulting currents are often related to fermion masses, and thus negligible for the first generation. Furthermore, the effect in our observables of interest would be suppressed, while other channels would be more sensitive due to chiral enhancement, e.g., π → eν e /π → µν µ if a charged current were generated [72,73], and π 0 → e + e − for a neutral current [74,75]. The aim of our analysis is then to identify correlations among the processes described above, compare sensitivities for a given NP scenario, and project the reach of future measurements in the various search channels.
In order to analyze these simplified models we will first perform the matching onto SMEFT (thus explicitly respecting SU (2) L gauge invariance) in Sec. 2. Then in Sec. 3 we express the relevant observables in terms of the Wilson coefficients of SMEFT, including detailed discussions of the respective master formulae and NP constraints. The phenomenological analysis of the four classes of simplified models is performed in Sec. 4, before we conclude in Sec. 5.

Standard Model Effective Theory and Simplified Models
As motivated in the introduction, we consider four different classes of new particles that modify quark-lepton interactions at tree level: LQs, VBs, VLQs, and VLLs. As we assume the NP to be heavy, i.e., to be realized above the EW symmetry breaking scale, we can match these models onto SMEFT. This makes SU (2) L gauge invariance explicit, and allows for a straightforward comparison of the additional relations that arise in each simplified model. In this section, we will first establish our conventions for the SMEFT operator basis, before defining the NP models and performing the matching.

SMEFT
Following the conventions of Ref. [70], we use the Lagrangian where the chirality conserving dimension-6 operators (we do not consider scalar or tensor operators here) that generate 4-fermion contact interactions between quarks and leptons are

2)
analysis of β decays and LHC bounds. while the operators generating modified gauge-boson couplings to fermions are Here q and are the quark and lepton SU (2) L doublets, while u, d, and e are SU (2) L singlets. φ is the Higgs doublet, D µ the covariant derivative, and τ I are the Pauli matrices. As we only consider first-generation fermions in this article, we can set i, j, k, l = 1 and omit the flavor indices in the following. In general, if left-handed quarks are involved, flavor-violatingūc and/ords couplings are generated after EW symmetry breaking, which lead to various effects in flavor observables. However, by assuming alignment to the down basis, down-quark flavor-changing neutral current are avoided and only D 0 -D 0 mixing remains as a relevant constraint. Here the SM contributions cannot be reliably calculated, in such a way that the bounds can always be avoided if one allows for a certain degree of cancellation between SM and potential NP contributions, and we will therefore not consider flavor bounds in the following.

Simplified Models and SMEFT
Extending the SM by LQs, VBs, VLQs, or VLLs gives rise to modified axial-vector or vector quark-lepton interactions described by the effective Lagrangian in Eq. (2.1) at tree level [71]. Here, we define LQs via their coupling to SM fermions, i.e., they have a vertex involving both a lepton and a quark. VBs are understood as QCD neutral spin-1 particles and VLQs (VLLs) are SU (3) triplets (singlets) that can couple to SM quarks (leptons) via the Higgs in an SU (2) L invariant way.

Leptoquarks
Ten LQ representations exist [76], of which five are scalars (Φ) and five are vectors (V ) as given in Table 1. Some of the LQ representations can have multiple couplings to quarks and Table 2. Couplings of the ten LQ representations to the SM quarks and leptons in the Lagrangian. 14 different couplings are possible, the ones of the scalar LQs are denoted by λ, the ones of the vector LQs by κ.
leptons, such that overall 14 gauge-invariant interaction terms with quarks and leptons 2 are possible, as listed in Table 2. A complete set of LQ interactions can be found in Ref. [79]. The tree-level matching onto the dimension-6 4-fermion operators in Eqs. (2.2) and (2.3) then yields [80][81][82] where the lowercase m (capital M ) stands for the mass of the scalar (vector) LQs.

Vector Bosons
There are two possible representations under the SM gauge group for (QCD neutral) VBs that allow for couplings both to quarks and leptons: an SU (2) L singlet (triplet), denoted as Z µ (X I µ ), both with zero hypercharge. Then the possible interactions with fermions are (2.6)

Vector-like Quarks
There are seven possible representations of VLQs under SU (3) × SU (2) L × U (1) Y , as given in Table 3. The Lagrangian describing their interactions with SM quarks and the Higgs field is Integrating out these new particles at tree level gives where we have further assumed the limit of vanishing first-generation quark Yukawa couplings [83].

Vector-like Leptons
There are six representations of VLLs under the SM gauge group, as given in Table 4, which couple to SM leptons and the Higgs field via (2.10)

Observables
In this section we summarize the observables relevant for constraining our simplified models.

Parity-Violating Electron Scattering
Limits on NP couplings to electrons and first-generation quarks can be extracted from data on PVES off the proton and off nuclei. In this subsection, we review the corresponding theoretical expressions, the constraints that are currently available, and future prospects.

Low-Energy Scattering
Interference between electromagnetic and weak scattering amplitudes leads to a PV asymmetry A N e that can be measured with a longitudinally polarized electron beam incident on an unpolarized nucleon target where σ ± represents the cross section of the helicity-dependent elastic scattering, t = −Q 2 the four-momentum transfer squared, and the kinematic quantities are defined as with scattering angle θ. The quantities A N V /A (t) represent the asymmetries arising from the terms in which the vector/axial-vector part of the weak current appears on the quark side, commonly parameterized in the effective Lagrangian where C e 1q , C e 2q contribute to A N V,A (t), respectively. Writing including radiative corrections as detailed in Refs. [87,88]. The NP contributions, expressed in terms of the SMEFT Wilson coefficients defined in Eqs. (2.2) and (2.3), are given by where s W = sin θ W is short for the weak mixing angle. Next, the nucleon matrix elements are expressed in terms of form factors according to where q = p − p, t = q 2 . In particular, we will write F N i (t) for the electromagnetic form factors, G 3 .27641(56) the axial-vector coupling of the nucleon [89]), and define the Sachs form factors In these conventions, the asymmetries become where is an isospin-breaking correction [90] (while isospin breaking in the strangeness contribution has been ignored). The weak charge of the proton is then identified as but its SM prediction includes further radiative corrections not yet included in Eq. (3.5), leading to This value is slightly smaller than the naive application of Eq. (3.5), Q p w = 0.0714, and slightly larger than the reference value quoted in Ref. [91], Q p w = 0.0708(3), with a difference that traces back to a small change in C e,SM 1u [88]. The adjustments in Eq. (3.12) include part of the γZ box correction from Ref. [92].
The present best measurement of Q p w comes from the Q weak experiment [91,93,94] at Jefferson Lab, which measured the asymmetry at Q 2 = 0.0248 GeV 2 and θ = 7.90 • , yielding [91] A p e = −226.5(9.3) × 10 −9 . (3.13) The data were analyzed setting all Wilson coefficients except for Q p w to their SM values, which, at tree level, implies in agreement with Ref. [91] (note that our G 3 A (0) = g A > 0 has the opposite sign). Our formulation in terms of Wilson coefficients (3.9) automatically accounts for the relevant short-range radiative corrections, including what is called the "one-quark" axial-vector contribution in Refs. [95,96].
Updating the strangeness form factor using µ s = −0.017(4), r 2 M,s = −0.015(9) fm 2 , r 2 E,s = −0.0048(6) fm 2 [97], Λ A = 1.0(2) GeV (corresponding to the axial radius from Ref. [98]) for the dipole scale in G 3 A (t), and estimating the axial-vector couplings as g u,p A + g d,p A = 0.40 (5) where we have followed the same prescription for the energy-dependent part of the γZ box correction [92,[102][103][104][105] as in Ref. [91]. This value is perfectly in line both with the Q weakonly result from Ref. [91], Q p w = 0.0706 (47), and the combination with other PVES data, Q p w = 0.0719 (45). In our analysis, we will retain the complete master formula (3.9), as the subleading terms produce some sensitivity to combinations of Wilson coefficients other than those contained in Q p w . The uncertainty is currently dominated by experiment, with theory uncertainties estimated in Ref. [91] at the level of 4.5 × 10 −9 when expressed in terms of A p e . In the future, the measurement of Q p w will improve considerably by the forthcoming high-precision P2 experiment at the MESA accelerator in Mainz [106]. Conducting the experiment at lower momentum transfer ( Q 2 = 0.006 GeV 2 , θ = 35 • ) to reduce the size of the γZ box corrections, P2 aims to measure the proton weak charge with a relative precision of 1.83%, a more than three-fold improvement over Q weak . At this level of precision also theory input requires further scrutiny, see the discussion in Ref. [106], including the role of G u,d E/M (t) and further long-range corrections [107,108], such as PV γγ boxes involving a nucleon anapole moment (called "many-quark" contribution in Refs. [95,96]). With a dedicated backwardangle measurement planned to constrain the latter, the remaining uncertainty from nucleon form factors is projected more than a factor of two below the experimental uncertainties.
Finally, PVES scattering can also be measured off nuclei, but so far results are restricted to 208 Pb [109,110]. Future plans include 48 Ca [111] and 12 C [106], but in both cases the major motivation concerns the presently poorly understood neutron distribution in the nucleus, in such a way that it is unlikely that meaningful constraints on NP can be extracted from PVES off nuclei alone. However, in a similar way CEνNS is also sensitive to a combination of NP couplings and nuclear structure (see Sec. 3.2), so that improved NP constraints are expected from a combined analysis of both classes of measurements.

Atomic Parity Violation
Apart from the very challenging measurements of PVES off nuclei at low momentum transfer, nuclear weak charges can also be accessed in APV, exploiting asymmetry amplification by stimulated emission in a highly forbidden atomic transition. Experimentally, the ratio of the PV amplitude over the Stark vector transition polarizability β is measured (with the most precise results currently available for 133 Cs [112,113]), which then needs to be combined with atomic-theory calculations and independent input for β. The latter can be determined semi-empirically either via a measurement together with another hyperfine amplitude [114,115] or the scalar polarizability [116][117][118], leading to the recommendation β = 27.064 (25) 88] in units of the Bohr radius a B , but the uncertainty of this average does not include an error inflation to account for the 2.7σ tension between the two methods. Similarly, the coefficient of the atomic structure calculation has been under debate in the literature [119][120][121][122][123][124], with Ref. [88] recommending 0.8977(40) from Ref. [124], which is in 1.7σ tension with the more recent 0.8893 (27) from Ref. [125]. Finally, despite the amplified asymmetry in the atomic system, the result is still sensitive to nuclear structure input, i.e., the neutron distribution in the nucleus. In Ref. [126] the recent PREX-2 measurement [110] of PVES off 208 Pb, in combination with a correlation to 133 Cs established based on density-functional methods, was used to improve this aspect of the extraction of the weak charge of 133 (5), (3.17) but the pull goes into the opposite direction. In our analysis, we will use Eq. (3.16), bearing in mind that the uncertainties might be slightly underestimated.
The above discussion shows that the current 0.6% precision of the weak charge of 133 Cs is becoming limited by theory, indicating that future improvements are difficult in the Cs system. However, the PV effect is enhanced by another factor of 50 in Ra + atoms, with a Ra-based experiment under development at TRIµP [127] and ISOLDE [128]. The projected gain in sensitivity on s 2 W by a factor of 5 would correspond to a 0.1% measurement of the weak charge of Ra.

Parity-Violating Deep Inelastic Scattering
PVES can also be studied in deep inelastic reactions, with the master formula [129] which depends on various parton distribution functions contained in R C (charm), R S (strange), R V (valence), as well as a kinematic factor Y 3 . In contrast to low-energy scattering, the PVDIS process is also strongly sensitive to the C e 2q couplings for sufficiently large Y 3 . The most precise measurements come from the Jefferson Lab PVDIS collaboration, who measured PVDIS off a liquid deuterium target for two kinematic settings [129,130] A (1) with corresponding SM predictions A

Coherent Elastic Neutrino-Nucleus Scattering
Low-energy PV is also accessible in CEνNS, in which a neutrino interacts with a nucleus via a neutral current, and the elastic recoil of the nucleus is measured. This rare process was measured for the first time by the COHERENT experiment [132,133], and future measurements at a number of experiments worldwide are expected to provide complementary constraints on the Wilson coefficients probed in electron scattering. In analogy to Eqs. (3.3) and (3.4), we write the relevant effective Lagrangian as Radiative corrections to the SM values have been studied in Refs. [87,134,135]. Following the conventions from Ref. [87], one has where we included the flavor dependence of C ν ,SM

1q
. Since the additional corrections from γZ boxes and renormalization of the axial-vector current as in Eq. (3.17) are absent for CEνNS, this leads to the flavor-dependent weak charges  [135]. The main difference between Refs. [87,135] concerns the treatment of the non-perturbative uncertainties arising in γ-Z mixing diagrams involving light quark loops, but this only affects Q ν ,p w and thus Q ν w at well below the percent level. In either case, due to process-dependent corrections being absorbed into the definition, there is no direct correspondence to the weak charges as defined in electron scattering, in such a way that the model-independent comparison of NP constraints has to proceed in terms of the respective Wilson coefficients.
The cross section for CEνNS takes the form [136] dσ where E ν /E ν is the energy of the incoming/outgoing neutrino, T = E ν − E ν the nuclear recoil, q the momentum transfer, and m A the target mass. The main sensitivity to NP effects proceeds via the weak charge, which enters as a normalization for the quark vector operators, while the correction from axial-vector operators, which contribute for non-spinzero nuclei, is sensitive to C ν 2q . The respective nuclear form factors F w (q 2 ) and F A (q 2 ) are discussed in detail in Ref. [136]. The weak form factor is normalized as F w (0) = 1, but the momentum dependence is, in general, still sensitive to the C ν 1q , i.e., the short-distance physics does not simply factorize into the weak charge Q ν w . The axial-vector form factor decomposes as where J is the spin of the nucleus, the Wilson coefficients factorize into 29) and the nuclear form factors S T ij are defined in such a way that the one-body contribution is normalized to S n measurements need to be performed at very small momentum transfer, or improved nuclearstructure calculations are required, whose most uncertain part, again related to the neutron distribution in the nucleus, could be constrained in future global analyses of CEνNS and PVES off multiple nuclear targets. At present, ab-initio calculations are becoming available for medium-size nuclei [137], while currently most calculations are based on (relativistic) mean field methods [138][139][140][141][142] and the large-scale nuclear shell model [136,[143][144][145].
The NP constraints that can be derived from the CEνNS measurements by COHER-ENT for CsI [132] and Ar [133] are not yet competitive, see, e.g., Refs. [146,147], but future improvements are projected to even become interesting probes of the weak mixing angle [148,149]. In our analysis, we will thus first use the present COHERENT constraints, whose neutrino beam, through the decay of the π + , consist of a mixture of ν µ , ν e , andν µ . While flavor oscillations can be neglected on the scale of the experiment, this implies that only about a third of the events are actually sensitive to the first-generation NP operators, which will be taken into account in the respective exclusion plots. The current uncertainties on the total cross section, both for the CsI and the Ar measurement, are at the level of 30%, but in view of the broad CEνNS program worldwide, we also consider an optimistic projection of 1%-precision for a COHERENT-like decomposition of the neutrino beam. At this level of precision the subtleties regarding radiative corrections discussed above will also become relevant.

Beta Decays
In addition to the neutral current probed in electron and neutrino scattering off quarks, the same effective operators also contribute to β decays via SU (2) L invariance. In this case, observables depend on the CKM matrix element V ud , so to be able to extract NP constraints other processes that, indirectly, enter the determination of CKM parameters become important. Here, we briefly review the relevant processes and summarize the input we will use. Superallowed β decays remain the primary source of information on V ud [150]. While the data base, an average over a large number of nuclear decays, has been stable for many years, the accuracy of the extraction of V ud critically depends on hadronic [3,[151][152][153][154][155][156] and nuclear [150,[157][158][159] corrections. The former can be addressed in combination with lattice-QCD calculations [3], but improved control of the nuclear uncertainties will require a concerted effort that exploits recent advances in ab-initio nuclear-structure calculations [160][161][162]. Alternatively, V ud can be extracted from neutron decay [163,164], in which case the additional nuclear uncertainties are absent, leaving the same hadronic effects as for superallowed β decays. In this case, significant improvements in precision measurements of the neutron life time τ n [165,166] and the asymmetry parameter λ [89] have been achieved in the last years, so that, with Ref. [166] for τ n , a gain of another factor 2 in precision in λ would render the neutron-decay extraction competitive with superallowed β decays. Finally, pion β decay π ± → π 0 e ± ν e [167][168][169] is currently not measured sufficiently precisely to impose meaningful constraints, but might become relevant in the future [170] in particular in combination with K 3 decays [169].
The confrontation of the resulting value for V ud with CKM unitarity is then further complicated by a tension between K 2 and K 3 decays, leading to contradicting values for V us . For K 2 decays, traditionally, the ratio to the pion decay is studied, in such a way that only the ratio of decay constants [171][172][173] and radiative corrections [174,175] are required for the interpretation. Likewise, K 3 decays require input for the hadronic form factor f + (0) [176,177], but in contrast to K 2 radiative corrections [178][179][180][181][182] have not yet been independently verified in lattice QCD. Finally, also τ decays [36] allow for extractions of V us , which, however, do not resolve the K 2 and K 3 tension either.
In the end, this leads to a situation in which two tensions are present in a combined analysis of V us and V ud from kaon and β decays, one between K 2 and K 3 , the other with CKM unitarity. The exact significance depends on the choice for the various corrections described above, see Ref. [6] for a detailed analysis of the numerics, but a significance around 3σ should give a realistic estimate of the current situation. For definiteness we use the combination of Ref. [88] for the test of CKM unitarity, where they quote: while the tensions among the kaon decays cannot be related to the first-generation NP operators we study here. In fact, even though the deficit in the first-column CKM unitarity is less significant than the one of the first row, it does support an interpretation in which NP affecting V ud could lead to an apparent violation of CKM unitarity. In view of the higher precision we will use the constraint from the first-row unitarity test in the numerical analysis.

Electroweak Precision Observables
EWPO include Z decays measured at LEP as well as the W mass (LEP+Tevatron+LHC), the Fermi constant G F , the fine structure constant α, and its running from the low to the EW scale, which depends on ∆α had . 3 In this article we use the Python code smelli [192][193][194] v2.2.0 [195] to perform a global fit to the EWPO (a complete list is given in Table 5), in particular to constrain the effect of NP that modifies Z and W couplings to quarks and leptons. Note that among the set of EWPO two measurements deviate from their SM predictions by more than 2σ: the Z →bb forward-backward asymmetry A 0,b FB and the Z → e + e − asymmetry parameter A e . Since this later one involves first-generation fermions, it is of particular interest to our analysis, along with the ratio of the Z hadronic width to e + e − or µ + µ − pairs (R 0 e,µ ), which also deviates from the SM, albeit with small significance of about 1.4σ. Together this leads to a slight preference for non-zero NP effects for some simplified models in our phenomenological analysis.

LHC Bounds
Both the ATLAS and CMS collaboration recently presented a new non-resonant di-lepton analysis in Ref. [209] and Ref. [7], respectively. These data can be used to constrain the LQ and VB models as they give effects in non-resonant di-lepton events (assuming that the Z is heavier than ≈ 5.5 TeV). To this end, we first compute the partonic cross sectionŝ with q ∈ {u, d, s, c, b} at tree level. The SM tree-level amplitudes are with the parton center-of-mass energyŝ equal to the square of the invariant mass of the electron-positron pairs m 2 eē and the differential parton-anti-parton luminosities [210] given by where s is the beam center-of-mass energy and τ =ŝ/s. In our numerical analysis, we use the PDF set NNPDF23LO, also employed, e.g., in the ATLAS analysis, to generate the signal Drell-Yan process [209], with the help of the Mathematica package ManeParse [211].
The CMS collaboration also provided results for the differential cross-section ratio for the nine m ¯ ( = e, µ) bins between 200, 300, 400, 500, 700, 900, 1250, 1600, 2000, and 3500 GeV. By distinguishing the cases where zero, or at least one, of the leptons were detected in the CMS endcaps, eighteen values were obtained, denoted by R Data µμ/eē,i with i = 1, . . . , 18. These were then normalized to the SM predictions obtained from Monte- In the double ratio, many of the experimental uncertainties cancel [212] and, importantly, the double ratio was further normalized to unity in the first bin from 200 to 400 GeV to correct for the different detector sensitivity to electrons and muons. In order to relate the NP Wilson coefficients to the CMS measurements, we compute R SM+NP µμ/eē,i . We then determine the best fit to data via a χ 2 statistical analysis, defining where σ i are the experimental uncertainties reported in Ref. [7]. Here, CMS reported an excess in electrons of about 4σ, and taking into account the q 2 distribution, one can improve the description of the data by more than 3σ with respect to the SM [213]. ATLAS, on the other hand, did not provide the muon vs. electron ratio, but rather quoted limits on NP from non-resonant di-electrons in the signal region [2.2, 6.0] TeV ([2.7, 6.0] TeV) for the cases where the NP contribution is interfering constructively (destructively) with the SM. Even though their limits agree with the SM expectation within 2σ, slightly more electrons than expected are observed. Here, we have to recast the bounds on the Wilson coefficients (similarly to the method described for the CMS analysis), since ATLAS studied only operators that have equal couplings to up and down quarks, which is generally not the case for the simplified models we consider.
Concerning LQs, which contribute in the t-channel, we included the propagator according to the prescription in Ref. [214]. For Z bosons we did not include resonant LHC searches. This means implicitly that we assumed that M is above the production threshold, in such a way that the bound on M for g = 1, to be derived in the next section, should be understood as a limit on g/M .

Phenomenology
We now turn to the phenomenological analysis, in which we will again consider the different classes of new particles separately. First we will collect the bounds for each representation, and then have a more detailed look at low-energy PV including a discussion of future prospects.

ATLAS destructive ATLAS constructive
−0.9 9.9 Table 7. Lower limits on the LQ masses for |λ| = |κ| = 1. For the ATLAS di-electron searches [209] and PV the 95% C.L. limits are given, while for CMS [7] and the CAA the preferred 1σ regions are shown (if the representation improves the fit with respect to the SM). Scenarios that worsen the agreement with data compared to the SM are indicated by − and * marks if the corresponding observable is unaffected.

Leptoquarks
First-generation LQs were studied in Refs. [82,[214][215][216][217][218]. The summary of the bounds and preferred values for the LQ masses from LHC searches, PV, and the CAA is given in Table 7 and shown in Fig. 1 in the C e 1u -C e 1d plane. One can see that the ATLAS bounds from non-resonant di-lepton searches are very stringent, but still allow for an explanation of the CMS excess for the LQ representations with the couplingsλ 1 , λ LR 2 , λ RL 2 , λ 3 ,κ 1 , κ RL 2 , and κ 3 that give rise to constructive interference with the SM. While an explanation of the CAA with κ 3 is in tension with the ATLAS and APV+Q weak measurements, the scalar triplet with the coupling λ 3 could address the anomaly. 4 While the current limits from PV experiments are only in some cases competitive with, or superior to, LHC limits, the experimental prospects look more promising. In fact, the Ra + and P2 experiments would be able to distinguish different representations and, e.g., favor λ LR 2 and κ RL 2 as solutions to the CMS excess, if the current central values for C e 1u and C e 2u were confirmed. Regarding the PV experiments involving neutrinos, not even the 1% accuracy projection for an Ar-COHERENT-type experiment can compete with the direct search limits, partly because only the fraction f e ≈ 30% of electron neutrinos in the overall neutrino flux can be used to search for first-generation LQs.  Table 8. Lower limits on the masses of VBs for |g e | = |g q | = 1 extracted from the ATLAS dielectron analysis and PV, as well as the mass ranges preferred by the CAA and CMS data (1σ). The * mark indicates if the corresponding observable is unaffected. The signs in the brackets denote the sign of the product g e g q . The notation for the couplings is as in Eq. (2.5), with only the couplings as indicated in brackets in the first column non-vanishing.

Vector Bosons
For VB we consider two cases, an SU (2) L singlet (Z ) and an SU (2) L triplet (X I ). The Z couplings to left-and right-handed quarks and leptons are all independent, 5 and therefore it is evident from Eq. (2.6) that various combinations of Wilson coefficients can be generated.
In our phenomenological analysis we study the benchmark scenarios given in Table 8, which are shown in the C e 1u -C e 1d plane in Fig. 2. Even though the ATLAS bounds are more stringent in case of constructive interference with the SM, allowing only masses above (5-7)TeV (if the relevant BSM couplings are fixed to unity), we find that each scenario is able to explain the CMS excess at the 1σ level. While the Z does not affect the CAA, X I generates a contribution to C (3) q , which can, for constructive interference, explain the CAA and the CMS excess simultaneously (see Fig. 2). Concerning PV experiments, the current APV and Q weak measurements yield already competitive bounds for some cases, which would improve significantly with P2 and APV with Ra + . Assuming the central value to be the same as from current APV and Q weak experiments, one could, e.g., disfavor X I and only two scenarios for the Z would be preferred.

Vector-like Quarks
We first note that only the three representations U , D, or Q 1 (with couplings to both up and down quarks) can explain the CAA [1,[63][64][65] due to their mixing with the SM quarks, while, on the other hand, T 1 or T 2 worsen the CAA. EWPO places multi-TeV mass limits   Table 9. Ranges or lower limits for the masses of VLQ for |ξ| = 1 extracted from EWPO and PV (95% C.L.). The first two representations can explain the CAA at the 1σ level for the given mass rage, while − indicates that the fit is worsened with respect to the SM, and * denotes that the CAA is not affected.
on the VLQs U , D, and Q 1 (for couplings fixed to unity), which come close to ruling out the best-fit regions from the CAA for U and D, while there is actually a small (1σ) preference for the other representations (Q 5,7 and T 1,2 ) driven by R 0 e and R 0 µ . All these bounds are collected in Table 9 and shown in Fig. 3. While there have been several direct searches for VLQs coupling to first-generation quarks, the mass limits are ∼ 1 TeV [226][227][228][229] (or perhaps as high as ∼ 1.4 TeV in certain regions of parameter space for D or T 2 ) and so are currently weaker than any of the other observables considered in this paper.
Looking back at Table 9 and Fig. 3 we also see that Q weak and APV currently provide in several cases even better bounds than EWPO. Since the Q 1 VLQ has two free parameters, as it can couple to both up and down quarks, we show the full parameter space in Fig. 4. Concerning future prospects, a 1% accuracy is needed for an Ar-COHERENT-type experiment to be relevant, but P2 and Ra + would (assuming that the current central value is confirmed) still allow the CAA to be explained by either the D VLQ or a Q 1 VLQ coupling to both u and d quarks.

Vector-like Leptons
VLLs coupling to first-generation leptons are strongly constrained by EWPO from LEP and the LHC [84][85][86][230][231][232][233][234] as well as direct searches [235][236][237]. The direct search bounds are ∼ 700 GeV, ∼ 450 GeV, and ∼ 150 GeV for doublets, triplets, and singlets respectively, while EWPO bounds are even stronger: doublets and triplets to masses above ∼ 5 TeV and ∼ 4 TeV respectively, while there is some preference for the SU (2) L singlet VLL at around 7 TeV (for a coupling fixed to unity), driven by the discrepancy in R 0 e , R 0 µ , and A e . They also cannot substantially improve the CAA, since only V us as determined from K 3 decays is affected, while all the other main determinations of V us and V ud are unchanged with only first-generation lepton couplings (see Refs. [4-6, 85, 86, 238] for VLLs explaining the CAA with coupling to different lepton generations.). Furthermore, the EWPO bounds are   so strong that PV cannot compete here, even taking into account possible future improvements.

Limits and Prospects from Low-Energy Parity Violation
NP limits from low-energy PV have already been formulated in terms of the NP scale Λ for benchmark models in Refs. [91,129]. Here, we compare our results to the presentation in Ref. [91], where the combined limits from Q weak and APV are given as a function of the angle θ h = arctan C e,NP 1d /C e,NP 1u , as shown in Fig. 5. The first conclusion from the simplified-model analysis is that only discrete values in θ h arise, as indicated by the vertical lines in Fig. 5. Depending on the angle, the limits on the respective LQ masses can vary substantially, between 2.5 TeV and 10 TeV, see also Table 7, as reflects the fact that the different simplified models define a different trajectory in the C e 1u -C e 1d plane and thus a different intersection with the current exclusion ellipse, see Fig. 1. The projection for P2 + Ra + then emphasizes the discriminating power of the combined analysis: if the central values remained at the current APV + Q weak best-fit point, but errors were reduced as projected, the combined data would prefer the green region in Fig. 5, and thus eliminate most of the simplified models as suitable candidates.
The analysis in our paper is focused on the C 1q coefficients, but we do keep the full dependence also on C 2q (in the case of Q weak ) and studied the impact of the PVDIS constraints including projected limits from SoLID, see Sec. 3.1.3. Scenarios in which C 2q dominate are possible in Z models upon tuning the couplings accordingly [239,240], but the . The couplings λ LR 2 , κ LR 2 and λ 3 , κ 3 require a different normalization, therefore the different scales on the right apply when reading off the scale Λ (corresponding to the LQ mass). The black line refers to the current limit from APV + Q weak , the green region to the preferred space assuming P2 + Ra + precision and no change in the central values.
same coefficients can also be probed in the LHC di-lepton searches. As show in Ref. [241] in the framework of SMEFT, the estimated SoLID sensitivity to NP is at most of the same Figure 6. Graphical representation of the limits on the LQ and VB mass scale Λ for |λ| = |κ| = 1 and |g e | = |g q | = 1, respectively. Scenarios that worsen the agreement with CAA (CMS) data compared to the SM are indicated with black (yellow) minus signs next to the coupling label. The hatched regions indicate the P2 and Ra + exclusion prospects, assuming that the current central value remains unchanged. order as current LHC searches, and the simplified-model analysis yields the same conclusion. However, in the combination with P2 and di-lepton searches, it could be helpful in distinguishing different NP scenarios.

Summary and Conclusions
NP related to first-generation fermions is not only very constrained, but also particularly interesting due to the large variety of related observables, allowing one to probe many complementary aspects of possible SM extensions. More recently, the observed deficit in the first-row and -column CKM unitarity, known as the Cabibbo angle anomaly (CAA), as well as the excess in non-resonant di-electron searches observed by CMS, further motivate the study of NP effects in the first generation. Since, due to the large number of couplings in SMEFT, fully model-independent relations among different processes are still scarce, the identification of such correlations suggests the consideration of simplified models, especially the four classes that can give rise to modified 2-quark-2-lepton interactions at tree level (after EW symmetry breaking): leptoquarks (LQs), QCD-neutral vector bosons (VBs), vector-like quarks (VLQs), and vector-like leptons (VLLs).
After performing the matching of these models onto the dimension-6 SU (2) L gauge invariant SMEFT, we analyzed all relevant observables related to these operators, including β decays, EWPO, LHC bounds, and low-energy PV. In particular, we provided master formulae for PVES, Eq. (3.9), and CEνNS, Eq. (3.27), that express the observables directly in terms of the short-distance Wilson coefficients and the respective hadronic matrix elements. Our main results are summarized in Fig. 6 and Fig. 7, illustrating the complementarity of the different classes of observables. For instance, the LHC bounds obtained from the CMS and ATLAS non-resonant di-electron searches only apply to LQs and VBs, while these ex-tensions are in general poorly constrained by EWPO. VLLs are by far best bounded by EWPO, which is also the case for several VLQ representations. LQs, VBs, and VLQs can have a significant impact on PV, but the actual constraint depends on the representation, and, similarly, only a subset of the representations affects β decays and thus the CAA.
Concerning PV, different representations predict different trajectories in the C e 1q and C ν 1q planes departing from the SM point, which leads to different bounds given that the present exclusion limits are not symmetric. This is illustrated for LQs in Fig. 5, as a function of the phase θ h = arctan C e,NP 1d /C e,NP

1u
. Furthermore, future experimental improvements could allow one to disentangle different representations, most notably with next-generation PV experiments P2 and Ra + , as shown in Figs. 5, 6, and 7. In addition, we find that for VLQs larger effects in PV than in the other simplified NP models are possible, in such a way that future prospects for CEνNS also promise discriminating power.
Altogether, Figs. 6 and 7 show the complementarity of low-energy precision observables and LHC searches in constraining NP related to first-generation quarks and leptons. If, further, the CAA and/or the CMS excess in di-electrons were confirmed, PV and EWPO could be used to disentangle different NP models even in case the masses were so heavy that a direct discovery would require a center-of-mass energy only available at future colliders.