Neutrino scattering and B anomalies from hidden sector portals

We examine current constraints on and the future sensitivity to the strength of couplings between quarks and neutrinos in the presence of a form factor generated from loop effects of hidden sector particles that interact with quarks via new interactions. We consider models associated with either vector or scalar interactions of quarks and leptons generated by hidden sector dynamics. We study constraints on these models using data from coherent elastic neutrino-nucleus scattering and solar neutrino experiments and demonstrate how these new interactions may be discovered by utilizing the recoil spectra. We show that our framework can be naturally extended to explain the lepton universality violating neutral current B decay anomalies, and that in a model framework the constraints from neutrino scattering can have implications for these anomalies.


Introduction
In spite of the success of the standard model (SM) in describing the particle interactions observed in nature, neutrino interactions with matter are not thoroughly understood. Many experiments are now making precise measurements of neutrino-nucleus scattering cross sections and neutrino-electron elastic scattering cross sections. The measurements are now precise enough that they are able to probe beyond the SM physics. Recently, the COHER-ENT experiment at the Spallation Neutron Source (SNS) has measured coherent elastic neutrino-nucleus scattering (CEνNS) for the first time, finding that the cross section for scattering on CsI is consistent with the SM at approximately 1σ [1]. In addition, measurements by Borexino of the solar neutrino flux, in particular the 7 Be component, now provide the best measurement of the neutrino-electron elastic scattering cross section at electron recoil energies MeV [2].
Because of this plethora of current and future experimental data, it is imperative to consider new theoretical ideas for neutrino interactions in these low energy experiments. Consider axial-vector interactions between quarks and neutrinos mediated by a new Z boson. We write the interaction of the quarks with the Z as, L q q =q γ µ P L F L (q 2 ) + P R F R (q 2 ) q Z µ , (1.1) whereγ µ = γ µ − γ·q q µ q 2 and q are SM quark fields. The interaction of the Z with the leptons has an analogous expression. In the interaction above, the contribution from the q µ part may be suppressed by small masses or vanish from current conservation. Form factors proportional to σ µν q ν are possible, but will be suppressed by some hadronic scale; we do not investigate these in this paper.
The form factor F (q 2 ) can be unity when q couples directly to Z . However, in many models (especially the models with low scale hidden sectors), q may couple to Z via a loop containing DM/hidden sector particles. In such a scenario we expect F (q 2 ) ∼ q 2 /Λ 2 where Λ is the scale in the DM sector associated with the mass of the mediator particle that generates the quark-DM interactions,qqχχ. As long as Λ is greater than q max for these scattering experiments, F (q 2 ) ∼ q 2 /Λ 2 appears in the scattering amplitude. In this paper, we investigate these new form factors at CEνNS (COHERENT and reactor based) and Borexino experiments for vector and scalar mediators. We extend our framework to study the neutral current B decay anomalies in the R K and R K * measurements. We show how in a model framework measurements from neutrino scattering may have implications for the B anomalies.
The paper is organized as follows. In Section II, we discuss models for form factors and their dependence on the new physics scale. In Sections III and IV we discuss the effects of these new form factors at CEνNS and Borexino experiments, respectively. In Section V we discuss the B anomalies and the implications of neutrino scattering experiments on their explanations. We conclude in Section VI.

Z Model
Consider the following Lagrangian at low energy [3]: where χ is a hidden sector fermion field with mass m χ . The first term in the Lagrangian represents an effective coupling between the q and χ fields that might arise through the exchange of a heavy mediator of mass ∼ Λ, with Λ E, where E is the energy scale of the process (see for example [4][5][6]). The hidden sector fields χ couple directly to Z through the vector portal and so in our framework there are two mediators; see e.g., [7]. We further assume that the neutrinos are charged under the Z U (1) and so there is a direct coupling of the neutrinos to Z . Although there is no direct coupling between the quarks and Z field, a χ-loop-inducedqqZ effective vertex (as in Eq. 1.1) will be generated with a q 2 dependent coupling, which can be represented by a higher dimensional operator, where Z µν is the Z field tensor; see Fig. 1. This higher dimensional operator may be considered to be the bare term of the Lagrangian. The form factors in Eq. (1.1) are then given bȳ    The most general structure for the form factors from the current conservation of J µ,χ is .2) and (2.3), we can estimate the form factor in the 1-loop approximation [8].
We get . Introducing a cut-off Λ c to regulate the divergent integral we can write where g and g χ are the bare coupling constants. We rewrite g L,R (q 2 ) as and q 2 max is the maximum momentum transfer squared. Note that this is a rough estimate as we have calculated only the leading term in Eq. (2.3). However, the general structure of Eq. (2.4) still holds as it follows from vector current conservation. As a rough estimate, assuming all terms in Eq. (2.7) to be of the same size, we can write, Although the Lagrangian in Eq. (2.1) contains both g L and g R terms, only g L +g R ≡ g v contributes to ν-nucleus coherent scattering; g L − g R ≡ g a does not impact the scattering process. This is because the vector charge of the nucleus is proportional to the number of nucleons, Q v = Zg p v + N g n v while the axial vector couplings are proportional to the spin, Q a = g p a S p + g n a S n which is much smaller than Q v .

S Model
As for the Z case the form factors for a scalar mediator S can be written as where (2.10) In this case we cannot use current conservation. By calculating the leading loop contribution we find, (2.11) Again introducing a cut-off Λ c we can write, Unlike the Z case the form factor is sensitive to the χ mass and so the hidden sector dynamics can be probed in low energy scattering.
A scalar coupling has a larger effect on the coherent scattering rate ∼ O (E R ), compared to the rate from a pseudoscalar coupling, 1 Note that the scalar coupling may originate from the dimension-5 operator, φHl R l l /Λ, where φ is a SM scalar singlet and H is the SM Higgs doublet. If φ is a pseudoscalar, then the associated coupling is not constrained by coherent scattering but is constrained by π 0 →νν [9] to be smaller than 10 −5 .

Scattering cross sections
For the vector model we can write the effective interaction as where F q 2 , Λ 2 = q 2 Λ 2 . Since the vector interaction has the same structure as in the SM, its contribution can interfere with the SM contribution.
The neutrino-electron differential cross section can be written as for the electron neutrino, and L = − 1 2 + sin 2 θ w + gF (q 2 ,Λ 2 ) for the muon or tau neutrino.
The neutrino-nucleus differential cross section is where F nucl q 2 is the nuclear form factor, the "weak charge" Q V is given by We have assumed the Z couplings are the same for the up and down quark.
On the other hand, for the scalar model, the effective interaction is which does not interfere with the SM. We can write its contribution to the differential cross section as for electron scattering, and for nucleus scattering [10]. Here Z (N ) corresponds to the number of protons (neutrons).

Experimental bounds
This section describes the experiments that we use to bound the aforementioned models. We begin by discussing accelerator and reactor experiments, and then discuss solar neutrino experiments.

Accelerators and reactors
To evaluate current and future constraints from accelerator and reactor CEνNS experiments, we use a χ 2 analysis to calculate bounds on the coupling at the 2σ confidence level.
To take into account reactor and accelerator neutrino flux uncertainties, we introduce a nuisance parameter α and an uncertainty on the signal of σ α . We define where N i obs (N i th ) is the observed (predicted) number of events per bin in a current measurement, σ α = 0.28 and σ i stat is the statistical uncertainty which can be extracted from Ref. [1]. For future measurements with multiple detectors we define (with indices suppressed), where N SM is the expected number of events in the SM for a future measurement and N bg is the expected number of background events, which we assume to be known precisely. Here we estimate σ α = 0.1 for future measurement.
The current COHERENT experiment has a threshold 4.25 keV [1]. For the future projected measurements we assume a threshold of 100 eV for Ge and Si reactor experiments [11][12][13], and 2 keV for NaI and Ar with COHERENT [14]. For reactor neutrinos we take a background of 1 dru (Ge and Si), and for accelerator neutrino data we take a background of 5 × 10 −3 dru (CsI, NaI and Ar) [1]. Here the unit dru stands for differential rate unit, equal to event/ (keV · kg · day). The COHERENT experiment has an energy dependent efficiency. We applied the efficiency function from [1] to all the detectors in the COHERENT experiment. We take the reactor neutrino flux to be that of a 1 MW reactor at ∼ 1 m from the core (which yields a the total flux of 1.5 × 10 12 cm 2 /s), and the antineutrino fission spectrum at various sites from Ref. [15]. The accelerator neutrino flux at SNS is 4.29 × 10 9 cm 2 /s [1].
In Figs. 2 and 3 we show the COHERENT and reactor constraints on g q 2 Λ 2 = ((g L +g R )gν )q 2 0 2Λ 2 at 2σ for a vector or scalar mediator, respectively, as a function of the mediator mass. g q 2 Λ 2 represents the coupling strength between quarks and neutrinos as a function of energy and reduces to g if there is no form factor for the coupling. We choose q 0 to be a typical momentum for the experiment, e.g., q 0 = 50 MeV and 30 MeV are used for COHERENT and reactor experiments, respectively. To compare with the limits for the case without a  form factor, we plot the corresponding limits using dashed lines. The quarks may have direct couplings to the Z and may also couple via DM loops in a given model, in which case the solid and dashed lines must be combined to obtain constraints on the couplings. The plateau for small mediator masses arises because m 2 q 2 which makes the limits independent of the mediator mass. In the regime of large mediator masses, the slope of the limit curves is 2 since the effective couplings become g m 2 , i.e., log g ∝ 2 log m . Also notice that there is a bump in the low mass region for future COHERENT and reactor experiments because a combination of the form factor and the mediator propagator yields q 2 q 2 +m 2 ∼ 1, so that the mediator-induced spectral distortion is suppressed. On the other hand, for the case with no form factor, the shape distortion persists for low masses, which makes the limits stronger compared to the F q 2 ∼ q 2 case. Note that direct detection constraints are nonexistent for sub-GeV DM and collider bounds are nonexistent for a GeV mediator which allows a lot of the parameter space to be unconstrained for g ≤ 1.
An effect of the form factor, F (q 2 ) ∼ q 2 , is that the spectral shapes differ from the SM prediction and from new physics models with F (q 2 ) = 1. To illustrate this, we show the spectrum of coherent scattering off a Ar target in Fig. 4. We choose the coupling g from current COHERENT constraints for F (q 2 ) ∼ q 2 (solid line) and F (q 2 ) = 1 (dashed line). The main difference between the solid lines and dashed lines are at the higher energy end because the form factor q 2 enhances the deviation from the SM. At low energy, the spectrum is suppressed by the detection efficiency.

Solar neutrinos
Several solar neutrino experiments, for example Super-K [16], SNO [17], and Borexino [2], are sensitive to the neutrino-electron elastic scattering detection channel. Since the typical momentum transfer that solar neutrino experiments are sensitive to is ∼ 0.4 MeV, it is possible to probe much smaller values of Λ as compared to reactor and accelerator CEνNS experiments. Here we consider all the most prominent low energy components of the solar neutrino flux that Borexino is sensitive to, i.e., pp, pep, and 7 Be. We choose the high metallicity solar model as defined in Ref. [18] for our baseline Standard Solar Model (SSM), and comment on the impact of the model uncertainties below. For solar neutrino experiments, the systematic uncertainties dominate. So we define χ 2 for each component of the solar flux to be where σ is the percent uncertainty in the measurement (including experimental and theoretical uncertainties in quadrature) with σ pp = 0.11, σ7 Be = 0.03, and σ pep = 0.21 [2]. To obtain a combined limit we define χ 2 = χ 2 pp + χ 2 7 Be + χ 2 pep . In Figs. 5 and 6, we show the constraints on the eeνν coupling from Borexino [2]. We find that the pp and 7 Be components provide the strongest constraints on F (q 2 ) ∼ q 2 because of their higher event rates and smaller flux uncertainties. This is despite the fact that the pep component has larger spectral distortions (for the form-factor case relative to the F (q 2 ) = 1 case) due to its higher energy. The limit plots are valid as long as Λ 2 q 2 . As for the nucleus scattering case, the recoil spectra in Fig. 7 show that the F (q 2 ) ∼ q 2 case is different from the F (q 2 ) = 1 case. We see that the major differences in the spectra are  Figure 5. Constraints at 2σ from the Borexino experiment on a vector mediator with F q 2 ∼ q 2 as a function of the mediator mass, compared to the case of a mediator without a form factor (dashed line). We set q = 0.5 MeV and Λ = 10 MeV for the form factor case to compare it to the no-form-factor case.
at high energies. The differences for the scalar case are more significant than for the vector case because in the vector scenario the q 2 enhancement is suppressed by the interference between SM and new physics contributions.

B anomalies
In the SM the three families of quarks and leptons are identical except for their masses. Tests of the universality of leptonic interactions are crucial probes of new physics. Recently, hints of lepton universality violating (LUV) measurements in B decays have attracted a lot of attention. These anomalies are found in the charged current b → cτ −ν τ and neutral current b → s + − transitions. Here we focus on the neutral current anomalies though the anomalies might be related [19][20][21]. The LHCb Collaboration has measured the ratio   Figure 6. Constraints at 2σ from the Borexino experiment on a scalar mediator with F q 2 ∼ q 2 as a function of the mediator mass, compared to the case of a mediator without a form factor (dashed line). We set q = 0.5 MeV and Λ = 10 MeV for the form factor case to compare it to the no-form-factor case. observable in this case is R K ≡ B(B + → K + µ + µ − )/B(B + → K + e + e − ) [23,24], which was measured by LHCb [25]: and found to differ from the SM prediction, R SM K = 1 ± 0.01 [26] by 2.6σ. Other anomalies also appear in the branching ratios and angular observables of certain b → sµ + µ − decays. While many new physics models with new heavy states have been discussed to address these anomalies, it was pointed out that new physics with light mediators could also explain these anomalies [27]. In particular, with heavy new physics it is difficult to understand the R K * measurement in the low q 2 bin, 0.045 ≤ q 2 ≤ 1.1 GeV 2 , along with the measurement of R K * in the central q 2 bin and the measurement of R K .
For light new physics in the MeV range a resolution of the R K and R K * measurements in the central q 2 bin along with other angular observables in b → sµ + µ − decays is possible with the light states coupling only to muons [3,27,28]. In addition, in this framework the discrepancy in the anomalous magnetic moment of the muon can also be explained and there are interesting implications for nonstandard neutrino interactions. However, the measurement of R K * in the low q 2 bin cannot be satisfactorily explained. For the model to work a nontrivial form factor for the flavor changing bsX vertex is required, where X is a light state. This can happen if the bsX coupling is induced at loop level due to some additional light new physics [3] just as we have considered in the case of neutrino scattering. To explain the R K and R K * in all bins with a light mediator is difficult and requires X to couple to electrons rather than muons [3]. In this case the anomalies in the angular observables in b → sµ + µ − decays remain unexplained. This might suggest that there is different new physics responsible for measurements in different q 2 bins. One can also aim to understand only the low q 2 bin R K * measurement and such an approach is followed in Ref. [29].
It is possible to connect B decays to coherent neutrino scattering by generalizing Eq. (2.1) to include all generations of quarks. We write the modified Lagrangian as where i, j are the family indices and Y U,D are the flavor couplings for the up and down quarks. To simplify the discussion we assume that only the left-handed quarks are involved in the interactions with the χ fields. However, in order to satisfy the R K and R K * anomalies we need flavor violation in the b − s sector arising from the following Yukawa matrices: The origin of the texture in the Y U,D can be understood by introducing a new gauge symmetry motivated by a U (1) µ−τ model [30][31][32][33], and including the quark sector. We assume that the Lagrangian has a similar symmetry in the quark sector with the following new Yuakawa terms: λ d

1Q
(2) 3,4 have new gauge charges 2a, -2a respectively, in addition to the SM weak charge assignments (2, 1/2) under SU (2) L and U (1) Y . Similar terms for the up sector are present as well. Such a model has been constructed in Ref. [34]. Here we assume that the quarks transform as (0, a, −a) but we could have assumed (a, a, −2a) as well with different charge assignments for the new Higgs.
In the weak interaction basis, the couplings to Z associated with the new symmetry is diagonal, In transforming from the gauge basis to the mass basis (with the contributions arising from the off-diagonal terms in the Lagrangian), we write The Y U,D matrices then have the explicit form, where c D ≡ cos θ D and s D ≡ sin θ D . We see that in the down sector flavor changing b → s transitions occur with coupling g bs = c D s D (g 2 −g 3 ). The form factor for coherent scattering is F (q 2 ) = g L q 2 Λ 2 while for the B decays it is F (q 2 ) = g Lbs q 2 Λ 2 with g L ∝ g 1 , g Lbs ∝ g bs and g 1 g bs = g L g Lbs . If all the g i are of the same order of magnitude then g Lbs < g L . The breakdown of lepton flavor universality required for the R K ( * ) anomaly can arise in U (1) µ−τ symmetry models. We now compare the flavor violating terms with the flavor conserving terms in the quark sector.
Combining the B decay anomalies with the results from coherent scattering allows us to check for the consistency of this framework. We focus on the Z models. Figure 2 gives the bound on the diagonal term using the COHERENT experiment, where g µ is the Z coupling to muon neutrinos, and we can read off X l from the figure.
We now turn our attention to the R K ( * ) anomaly which involves a flavor violating b − s interaction with charged muons. Using the recent results on R K ( * ) , we obtain a constraint on the flavor violating term. We assume the left handed leptons have identical couplings and so g µ can be fixed from the muon anomalous magnetic moment measurement and neutrino trident production. Using g µ ∼ 10 −3 [3,28] and X l ∼ 10 −9 from Fig. 2 we obtain Λ > 10 3 q 0 g L , (5.10) which gives Λ > 50g L GeV for q 0 = 50 MeV. In B decays the relevant q 0 is taken to be m B and so with the additional assumption of SU (2) L symmetry for the left handed leptonic couplings we obtain [3], Combining this with Eq. (5.9), we get (5.12) Using X h ∼ 10 −8 [3] and X l ∼ 10 −9 from Fig. 8, we find g bs /g 1 ∼ 10 −3 for m between 1 − 10 MeV, and g bs /g 1 ∼ 10 −4 for m = 100 MeV. However, if the bounds from coherent scattering get stronger, then g Lbs g L will increase and lead to tension in the framework. A similar analysis can also be done with scalar mediators where tighter constraints are obtained. The B anomalies also indicate lepton universality violating new physics which will be interesting to check in neutrino scattering. For instance if the R K ( * ) anomalies are explained with mediators coupling differently to muons and electrons then ν µ and ν e scattering may show different new physics effects.

Conclusions
We have explored the limits of the effective couplings arising from a high energy scale (Λ) hidden sector associated with dark matter. We considered two general models which give rise to a coupling form factor that is proportional to the momentum q 2 . The Z model corresponds to vector couplings between neutrinos and quarks, and the S model corresponds to scalar couplings. At low energies, we have shown that it is possible to probe Λ via CEνNS experiments via the form factor which is induced by a DM (χ) loop. We considered scenarios in which Λ is ≥ 50 MeV, and in which the energy scale is ≤ 1 MeV. CEνNS experiments can probe the former case since Λ is higher than the momentum transfer but these experiments are unable to probe the latter case. To probe the scenario with small Λ, we used solar electron scattering experiments for which the momentum transfer is low.
In the Z model, COHERENT constrains the coupling to be ∼ 10 −8 at 2σ for small mediator masses. For large mediator masses, the bound scales according to log g ∝ 2 log m , as shown in Fig. 4.1. Atomic parity violation does better than most of the CEνNS experiments except those using reactor neutrinos. For small Λ the Borexino experiment puts 2σ constraints on the couplings O(10 −7 ) for a 100 MeV mass mediator. Since the momentum transfer is much smaller, the constraints scale like log g ∝ 2 log m as shown in Fig. 5.
In the S model, COHERENT constrains the coupling to be ∼ 10 −9 for small mediator masses. For large mediator masses, the bound scales according to log g ∝ 2 log m , as shown in Fig. 4.2. Atomic parity violation experiments do not constrain models with scalar mediators. For small Λ, the Borexino experiment puts 2σ constraints on the couplings O(10 −7 ) for a 100 MeV mass mediator. Since the momentum transfer is much smaller, the constraints scale like log g ∝ 2 log m as shown in Fig. 6.
Finally, we have extended our framework to quarks of all generations and have addressed the R K and R K * anomalies in rare B decays. We have shown that a resolution of the anomalies consistent with the present coherent scattering data is possible but future constraints from coherent scattering will provide stringent tests of the B anomalies explanation.