Borexino and General Neutrino Interactions

We derive constraints on all possible general neutrino-electron interactions (scalar, vector, pseudoscalar, axialvector and tensor) using the recent real time Borexino event rate measurements of pp, pep and 7Be solar neutrinos. The limits improve several previous ones from TEXONO and CHARM-II for incoming electron and muon neutrinos, and are the first ones for the tau flavor. Future improvements by next-generation solar neutrino experiments are also studied. The limits extend the physics reach of solar neutrino measurements to TeV-scale physics. Finally, the different properties of the new interactions for Dirac and Majorana neutrinos are discussed.


I. INTRODUCTION
Precision studies in neutrino physics allow to tighten the parameters of the standard 3-Majorana neutrino paradigm [1]. Indeed, the precision on the parameters of the PMNS matrix approaches the one of the corresponding CKM parameters, see e.g. [2]. Moreover, effects of new physics beyond the standard paradigm can be tested.
We will employ the Borexino measurements of low energy pp, pep and 7 Be solar neutrinos. As the originally produced electron neutrinos oscillate to muon and tau neutrinos, this allows to set limits on general interactions of all flavors. Previously, limits on general neutrino-electron interactions were obtained in Ref. [6] using TEXONO [30] and CHARM-II [31] data for electron and muon neutrinos, respectively. We will improve several of those limits, and set the very first ones on general tau neutrino interactions. Possible future limits by upcoming of hypothetical solar neutrino measurements are also estimated. If percent-level coupling strengths are measured, and the new interactions are interpreted in terms of new exchanged bosons, then new physics of weak and TeV scales is tested by solar neutrino experiments.
The paper is structured as follows. In Sec. II we set up the formalism of neutrino-electron scattering with general interactions. Section III describes the data and fit procedure that we follow, with the results being discussed in Sec. IV. We also address the differences between Dirac and Majorana neutrinos in this framework in Sec. V, conclusions are drawn in Sec. VI.

II. NEUTRINO-ELECTRON SCATTERING IN THE PRESENCE OF GENERAL NEUTRINO INTERACTIONS
In this section we lay out the general formalism to describe general neutrino interactions relevant for elastic neutrinoelectron scattering. Starting with the Standard Model (SM), we have neutral current (NC) and charged current (CC) interactions between the target electrons and the three flavors of neutrinos. To be specific, in the SM, ν e e-scattering involves the CC and NC interactions, while ν µ/τ e-scattering depends only on NC interactions. The effective SM Lagrangian for the NC interactions is given as where the vector and axial vector couplings are For the CC interactions, after a Fierz transformation one can write (flavor indices are suppressed) We are interested here in new neutrino physics that may show up in the form of general neutrino interactions. With this we denote new interactions for neutrino-electron scattering, that can be scalar, pseudoscalar, vector, axialvector or tensor. The effective four-fermion interaction Lagrangian is where Γ a ≡ I, iγ 5 , γ µ , γ µ γ 5 , σ µν ≡ i 2 [γ µ , γ ν ] are the five fermion operators, corresponding to Scalar (S), Pseudoscalar (P), Vector (V), Axialvector (A) and Tensor (T), respectively. Furthermore, following the convention in Ref. [6], we have i a = i for a = (S, P, T)and i a = 1 for a = (V, A). Including the factor i for the S, P, T interactions is necessary to have αα and˜ αα real. We assume that and˜ are hermitian matrices, i.e., αβ = * βα and˜ αβ =˜ * βα , so that Eq. (4) is self-conjugate 2 . Possible phases of the and˜ matrices are ignored in what follows.
For Majorana neutrinos some of the interactions in Eq. (4) cannot be written in terms of Majorana spinors. More specifically, in this case, the vector and tensor interactions with α = β for each generation should vanish (i.e., , which is a known property of Majorana spinors [3,16,32]. However, considering three generations of neutrinos, such interactions can still exist for α = β. Nevertheless, for Majorana neutrinos, the parameter space of the general Lorentz-invariant interactions is smaller than the one for Dirac neutrinos. Our analyses will be focused on Dirac neutrinos, we will address the difference to the Majorana case in Sec. V. Furthermore, we focus on flavor diagonal interactions, i.e. we constrain a αα and˜ a αα . The limits on the off-diagonal terms will be very similar. In general the new interactions of Eq. (4) are added to the SM interactions in Eqs. (1) and (3). The differential cross section of neutrino-electron scattering is found to be [6]: where m e is the electron mass, E ν is the neutrino energy and T is the electron recoil energy. The parameters A αβ , B αβ , C αβ and D αβ are defined as (given here for complex parameters and ignoring flavor indices for simplicity) To recover the explicit flavor indices, one only needs to add subscripts αβ to all quantities in Eq. (7); in addition one has Note that the SM couplings appear only in A, C and D. The term proportional to B is a pure new physics term that contains no SM contribution. We restrict our analysis to the total event rates. In this case the total cross sections in terms of the maximum recoil energy of electrons T max (E ν ) are the relevant observables. We can obtain the total cross section from Eq. (5) as where Here we have E ν < 0.420 MeV for the continous pp spectrum, and E ν = 0.862 MeV (1.44 MeV) for neutrinos from 7 Be (pep) reactions, respectively. For each spectrum we have reproduced with good agreement the expected event numbers quoted by Borexino in [9]. More details will be discussed in Sec. III B.
It is important to note that the term proportional to C in Eqs. (5) and (8) is suppressed by the kinematic factor proportional T E ν with respect to A. This naturally leads to a relatively tighter constraints on the parameters related to C. For antineutrinos A and C are replaced with each other in the cross sections.
As stated earlier, the cross sections in principle contain contributions both from flavor conserving and flavor violating processes. For simplicity, we will restrict ourselves to the flavor conserving case at the neutrino vertex, i.e. ν e e → ν e e and ν µ,τ e → ν µ,τ e scattering. As a consequence there are interference terms for the SM and new physics terms in the cross sections in Eqs. (5) and (8). Regarding those interference terms, note that there is no interference of the vector/axial terms with the scalar/pseudoscalar/tensor-type interaction terms. All such terms cancel out in the cross amplitude terms due to the products of the odd number of gamma matrices for vector/axial currents with the even number of gamma matrices in the scalar/pseudoscalar/tensor current. Thus, the scalar, pseudoscalar and tensor interactions are independent of the vector and axialvector currents and in particular do not interfere with the SM interactions. We will discuss this point in more detail in Sec. V.

III. SEARCHING FOR EXOTIC INTERACTIONS IN SOLAR NEUTRINO EXPERIMENTS
In this section we give details of the solar oscillation probabilities, event rate calculations and the statistical model used for our analysis.

A. Solar Neutrino Oscillation Probabilities
As solar neutrinos change their flavor from production to detection, we need to consider the survival probabilities for the pp, 7 Be and pep neutrinos that we will use for our model to fit with the data. We follow the notation from [33]. If there were no matter effects, the oscillation amplitude would be A αβ = U αi X i U † iβ , where i are mass indices while α and β are the flavor indices. Summation over the mass indices is implied. Here U is the neutrino mixing matrix and X is the diagonal phase matrix . Thus, the solar neutrinos oscillation probability would read Due to the very large distance between Sun and Earth we can take the averaged oscillation probability as Expressed in terms of mixing angles, the averaged probability for solar neutrinos is P ee = s 4 13 + (c 12 c 13 ) 4 + (s 12 c 13 ) 4 , where s ij ≡ sin θ ij and c ij ≡ cos θ ij in the commonly used notation [1].
Matter effects are important for precision studies, and depend on energy. Solar neutrinos from the low energy pp reaction, which has a continuous spectrum with energy E ν ≤ 0.420 MeV, witness very little matter effects. The flux event rate (phase-I) event rate (phase-II) Our prediction %age error (theo.) pp − 134 ± 10 +6 probability P ee has less than a percent difference from the path-averaged expression in Eq. (10). However, for the somewhat higher energy discrete spectra of 7 Be and pep neutrinos (E ν = 0.862 and E ν = 1.44 MeV, respectively), the matter effects are still small, up to 4-5%, but not entirely negligible. Therefore, we include the small modifications due to matter effects according to where is the effective mixing angle inside the Sun, N e is the electron number density at the center of the Sun, N res e = ∆m 2 12 cos 2θ 12 /(2E ν √ 2G F ) is the electron density in the resonance region, ∆m 2 12 is the solar mass-squared difference, θ 12 is the solar mixing angle and G F is the Fermi constant. For the continuous pp spectrum, we use the electron density at average pp production point in the above expressions and assume an exponential decrease of the density outward from the core in the analytic approximations as discussed in detail in Ref. [34]. This is an excellent approximation for r > 0.1R solar [35].
Taking the current best-fit values of the oscillation parameters [1], we find the vacuum value P vac ee = 0.558, which is modified to P pp ee = 0.554 for pp, P 7 Be ee = 0.536 for 7 Be and P pep ee = 0.529 for pep neutrinos in the case of matter effects.

B. Borexino, Event Rate Calculations and the χ 2 -model
We will consider five measurements made by the Borexino experiment since 2007 both in phase-I [7,8] and phase-II [9][10][11] runs. The pp spectrum was measured in phase-II only, 7 Be and pep spectra were measured in both phase-I and phase-II with an extensively purified scintillator in phase-II between December 2011 and May 2016 for total of 1291.51 days. The data obtained from the experiment is given in Table I.
For all five measurements we take the number of target electrons per 100 tons, N target e = 3.307 × 10 31 , as quoted in Ref. [11], while taking the pp reaction flux from Ref. [35] 3 . Since the 7 Be and pep fluxes have discrete spectra, we treat them as delta functions in our analysis to evaluate the rate in Eq. (13), see below. In addition, as done by Borexino in their analysis, we use the high-metallicity SSM flux values φ7 Be = 4.48 × 10 9 cm −2 s −1 at 0.862 MeV and φ pep = 1.44 × 10 8 cm −2 s −1 at 1.44 MeV in our calculations. In order to calculate the expected number of events in the Borexino detector, we can write down the expression for total rates as where P i ee is given in Eq. (11), with the index i indicating whether pp, 7 Be or pep neutrinos are considered. The cross sections σ e (E ν ) and σ µ,τ (E ν ) are given in Eq. (8). We include radiative corrections, which is an effect of about 2% [35]. Note that we assume in Eq. (13) equal fluxes of muon and tau neutrinos, which corresponds to maximal θ 23 . This will imply identical limits for the muon-and tau-flavor parameters. For non-maximal θ 23 there will in reality be slightly different different limits. For data fitting, we use the following χ 2 -estimator to constrain the parameters − → λ ≡ ( a ,˜ a ): where i runs over the solar neutrino sources pp, 7 Be and pep. In Eq. (14), R exp are the experimental event rates observed at Borexino in phase-I and phase-II with σ stat the statistical uncertainties for each of the five measurements, while R pre is the predicted event rate corresponding to each experiment, calculated using Eq. (13). The predicted and measured event rates are quoted in Table I, as well as our calculated values for comparison. We take the neutrino energy window of 100-420 keV for calculating the pp-neutrino event rate. The obtained results for the SM case are given in Table I. In Eq. (14), we also add a penatly term corresponding to each measurement to account for the theoretical uncertainities in the solar flux model for the three solar spectra and from the oscillation parameters, mostly from θ 12 since θ 13 and ∆m 2 12 are known very well. In Table I we quote the percentage uncertainties for each spectrum using Borexino's predicted event rates. We use the predicted percentage uncertainties as the constraints (σ α ) on the pull parameters (α i ). Since the five measured event rate values given in Table I are already background-subtracted we do not inlcude any background terms in our χ 2 -model. Additionally, since we are working with the event rates we are less affected by details of the detector energy resolutions or detector response, etc. As stated, since we are using a simple χ 2 -model for our analysis that is based on the total event rate analysis for each solar spectrum, and we consider only the statistical uncertainties, therefore, we are not not considering any systematic correlated or uncorrelated errors from the different measurements of Borexino. The statistical analysis we have implemented here has already been used for phenomenological new physics studies. For example, see Refs. [33,36] and several others. The validity of the χ 2 -model used here has been cross-checked for estimating the neutrino magnetic moments for the same data in Ref. [37]. The results of that work are in good agreement with those obtained by Borexino in Ref. [11] for phase-II data. As an explicit comparison, the analysis from Ref. [33] applied to phase-II data without CNO data gives for the Weinberg angle sin 2 θ W = 0.229 ± 0.038, to be compared to Borexino's result [38] of sin 2 θ W = 0.229 ± 0.026. The limits we will present in what follows are therefore conservative. We would emphasize at this point that full agreement between our results and Borexino's result [38] cannot be expected since we are not including the CNO data in our analysis as the direct rate measurement by Borexino is not available, we rather use data from phase I and phase II while Borexino uses only phase II data.

IV. RESULTS
Having described our fitting procedure, we present here the results. As the produced electron neutrinos oscillate also to muon and tau neutrinos, we study two scenarios: (i) new interactions appear only for ν e , and (ii) new interactions appear only for ν µ/τ (recall that we do not distinguish both flavors). Fig. 1 shows the result of our χ 2 -fit for the general interactions of electron neutrinos. The constraints are compared to previous constraints obtained in Ref. [6] using TEXONO reactor antineutrino data. Borexino improves the limits on V ee ,˜ V ee , A ee , T ee and˜ T ee . Fig. 2 shows the fit result for new physics acting only on muon/tau neutrinos. The constraints are compared to a previous constraint obtained in Ref. [6] using CHARM-II data. Borexino improves the limit on˜ V µµ . For a ττ and˜ a ττ these are the very first limits. The numerical values of the constraints in Figs. 1 and 2 are given in Tabs. II and III. It is also useful to give constraints on the parameters A, B, C and D that appear in the total cross section Eq. (8). The result is given in Fig. 3 and Tab. IV for electron and muon/tau neutrinos. We have performed here two-parameter fits setting the other two parameters to their SM values. The SM values of these parameters are given in the last two columns of Tab. IV, which in particular is B = 0. For future experiments (see Sec. IV A) we assumed the SM values of A, B, C and D.

A. Future Prospects from solar Data
There are several ideas floating around to further improve the precision on solar neutrino measurements with a precision of 1% or better. The main motivations behind these projects are the determination of the correct metallicity (low or high) solar model and photon fluxes from the Sun, a more stringent test of the LMA-MSW solution of the      4), for electron neutrinos. The black line is the limit obtained from Borexino event rates, the red line for hypothetical future measurements with event rate precision of 1%, see Sec. IV A. Indicated are the 1σ and 90% C.L. projections. The thick horizontal magenta line is the limit obtained from TEXONO data, taken from [6].     (4), for muon/tau neutrinos. The black line is the limit obtained from Borexino event rates, the red line for hypothetical future measurements with event rate precision of 1%, see Sec. IV A. Indicated are the 1σ and 90% C.L. projections. The thick horizontal magenta line is the limit on muon neutrino general interactions obtained from CHARM-II data, taken from [6].  neutrinos propagating through the solar matter and to explore exotic properties related to the solar neutrinos. One such project is the Jinping experiment [39]. In addition, future large scale dark matter direct detection experiments can provide precise solar neutrino measurements [40][41][42], ideas to use future long-baseline neutrino oscillation far detectors as solar neutrino experiments are also present [43].
As for different types of potential future experiments the precision of the individual solar neutrino sources is different, we conservatively adopt for simplicity that all the three low energy solar neutrino fluxes (pp, pep and 7 Be) will have been measured with a 1% precision and assume the SM values of the various ,˜ or A, B, C, D. With this projected precision we simulate our data and then fit all the parameters in a similar fashion as was done for the real data. The results of this analysis are displayed with red color distributions in Figs. 1 and Fig. 2 and with green color ellipses in Fig. 3.
As clear from Figs. 1 and Fig. 2, and can be read off from Tabs. II and III, the future solar data with 1% precision will improve the current bounds on non-standard vector, axialvector and tensor interactions by more than one order of magnitude while for the scalar and pseudoscalar ones they will improve by a factor of 3 to 5, in general. The future constraints at 90% C.L. on the parameters A, B, C and D are also shown with green color ellipses overlaid on the current constraints for comparison in Fig. 3.
This implies that for Majorana neutrinos, the vector and tensor interactions are flavor anti-symmetric; the scalar, pseudoscalar and axialvector interactions are flavor symmetric. Therefore,
Note that these and˜ matrices should also be hermitian -see footnote 2, so Eq. (23) is equivalent to Majorana : Re a αβ = Re˜ a αβ = 0, (for a = V, T) Im a αβ = Im˜ a αβ = 0 (for a = S, P, A) , which means a αβ and˜ a αβ are real symmetric matrices for a = S, P, and A, and imaginary anti-symmetric matrices for a = V, and T. In particular, the diagonal parts of the vector and tensor coupling matrices should vanish, V αα =˜ V αα = T αα =˜ T αα = 0.
In summary, the difference between Dirac and Majorana neutrinos in the framework of this paper is that the and˜ matrices for Majorana neutrinos are further constrained by Eq. (23), or equivalently, Eq. (24). Thus our results based on Dirac neutrinos are readily applicable to Majorana neutrinos except that some of the couplings, namely flavor-diagonal V and T , as well as˜ V and˜ T , should be absent.

VI. SUMMARY AND CONCLUSIONS
We have discussed here the sensitivity of Borexino to general neutrino interactions. Assuming the presence of additional scalar, pseudoscalar, vector, axialvector or tensor interactions we have investigated how Borexino's measurements of pp, pep and 7 Be neutrino event rates constrain the dimensionless (i.e. normalized to the Fermi constant) interaction strength of the new interactions. Several previous limits from TEXONO and CHARM-II are improved for the electron and muon sector, while first limits on tau sector interactions were set. Our limits are summarized in Figs. 1, 2 as well as Tabs. II and III. We focused on Dirac neutrinos, and detailed the difference to Majorana neutrinos. Future prospects on the limits were also considered. Interpreting the interaction strengths as due to some new exchanged boson with coupling g X and mass M X implies that or˜ is given approximately by (g 2 X /M 2 X )/G F . This means that current (future) solar neutrino experiments are sensitive to new physics of weak (TeV) scale and beyond.