$pp$ Solar Neutrinos at DARWIN

The DARWIN collaboration recently argued that DARWIN (DARk matter WImp search with liquid xenoN) can collect, via neutrino--electron scattering, a large, useful sample of solar $pp$-neutrinos, and measure their survival probability with sub-percent precision. We explore the physics potential of such a sample in more detail. We estimate that, with 300 ton-years of data, DARWIN can also measure, with the help of current solar neutrino data, the value of $\sin^2\theta_{13}$, with the potential to exclude $\sin^2\theta_{13}=0$ close to the three-sigma level. We explore in some detail how well DARWIN can constrain the existence of a new neutrino mass-eigenstate $\nu_4$ that is quasi-mass-degenerate with $\nu_1$ and find that DARWIN's sensitivity supersedes that of all current and near-future searches for new, very light neutrinos. In particular, DARWIN can test the hypothesis that $\nu_1$ is a pseudo-Dirac fermion as long as the induced mass-squared difference is larger than $10^{-13}$ eV$^2$, one order of magnitude more sensitive than existing constraints. Throughout, we allowed for the hypotheses that DARWIN is filled with natural xenon or $^{136}$Xe-depleted xenon.


I. INTRODUCTION
Multi-ton-scale, next-generation dark matter experiments are expected to collect significant statistics of atmospheric and solar neutrinos. The DARWIN collaboration recently argued that DARWIN (DARk matter WImp search with liquid xenoN) can collect a large, useful sample of solar pp-neutrinos, measured via elastic neutrino-electron scattering [1]. There, they argued that the survivial probability of pp-neutrinos can be measured with sub-percent precision and that one can measure the Weinberg angle at low momentum transfers with 10% precision, independent from the values of the neutrino oscillation parameters. Here, we explore other neutrino-physics-related information one can obtain from a high-statistics, high-precision measurement of the pp-neutrino flux.
In a nutshell, pp-neutrinos are produced in the solar core via proton-proton fusion: p + p → 2 H + e + + ν. The vast majority of neutrinos produced by the fusion cycle that powers our Sun are produced via proton-proton fusion. ppneutrinos have the lowest energy among all solar neutrino "types" (other types include pep-neutrinos, 7 Be-neutrinos, 8 B-neutrinos, and CNO-neutrinos) and are characterized by a continuous spectrum that peaks around 300 keV and terminates around 420 keV. Theoretically, the pp-neutrino flux is known at better than the percent level [2] given they are created early in the pp-fusion cycle -they are the first link in the chain -and their flux is highly correlated with the photon flux, measured with exquisite precision. For the sake of comparison, the flux of 7 Be-neutrinos and 8 B-neutrinos, which provide virtually all information on the particle-physics properties of solar neutrinos, can be computed at, approximately, the 6% and 12% level, respectively [3,4]. The pp-neutrino flux has been directly measured, independent from the other flux-types, by the Borexino collaboration [5], with 10% precision.
A percent-level measurement of the pp-neutrino flux is expected to be sensitive to new-physics effects in neutrino physics that are also at the percent level. This includes, for example, effects from the so-called reactor angle θ 13not new physics but very small for solar neutrinos -and the presence of new neutrino states or neutrino interactions. Furthermore, the fact that pp-neutrinos have energies that are significantly lower than those of the other solar neutrino types renders them especially well-suited to constrain (or discover) new, very long oscillation lengths associate to very small new neutrino mass-squared differences. These searches are expected to add significantly to our ability to test the hypothesis that the neutrinos are pseudo-Dirac fermions [6][7][8] (for relevant recent discussions, see, for example, [9][10][11][12]). Here, as far as new-physics hypotheses are concerned, we concentrate on the search for new, very light neutrino states.
In Sec. II, we review the relevant features of the proposed DARWIN experiment and provide information on how we simulate and analyze DARWIN data on pp-neutrinos. In Sec. III, we show that a percent-level measurement of the pp-neutrino flux allows for a "solar-neutrinos-only" measurement of sin 2 θ 13 . In Sec. IV, we compute the sensitivity of DARWIN to the hypothesis that there is a fourth neutrino with a mass m 4 that is quasi-degenerate with the mass of the first neutrino state, m 1 (in the Appendix, we discuss how this can be generalized). We concentrate on the region of parameter space where the new mass-squared difference is 10 −13 eV 2 |m 2 4 − m 2 1 | 10 −6 eV 2 . We add some concluding remarks in Sec. V.

II. DARWIN AS A LOW-ENERGY SOLAR NEUTRINO EXPERIMENT
DARWIN is projected to be a large -40 tons fiducial volume -liquid xenon time-projection chamber, aimed at searching for weakly interacting massive particles (WIMP) in the GeV to TeV mass range [13] via elastic WIMPnucleon scattering. It will inevitably be exposed to a large flux of solar and atmospheric neutrinos and is large enough that solar-neutrino scattering events will occur at an observable rate.
According to [1], DARWIN is expected to collect a sample of almost ten thousand pp-neutrinos per year via elastic neutrino-electron scattering: where α = e, µ, τ is the flavor of the incoming neutrino. The flavor of the outgoing neutrinos is, of course, never observed. For pp-neutrino energies, the cross section for ν e e-scattering is around six times larger than that of ν a escattering, a = τ, µ and the differences between the cross sections for ν µ e-scattering and ν τ e-scattering are negligible. At leading order in the weak interactions, the differential cross section in the rest frame of the electron is where T is the kinetic energy of the recoil electron, E ν is the incoming neutrino energy, m e is the electron mass and G F is the Fermi constant. The dimensionless couplings a α , b α are where θ W is the weak mixing angle. DARWIN measures the kinetic energy spectrum of the recoil electrons. If filled with natural xenon, one expects a large number of electron-events in the energy range of interest from the double-beta decays of 136 Xe. These events are a powerful source of background for solar-neutrino studies and, according to [1], obviate the study of solar neutrinos with energies higher than 1 MeV. They are a powerful nuisance for measurements of the 7 Be-neutrino flux and have a significant but not decisive impact on the measurement of the pp-neutrinos (around a 30% decrease in the precision with which the overall pp-neutrino flux can be measured [1]). The reason one can measure the pp-neutrino flux in spite of the 136 Xe background is that the shape of this particular background is well known and the experiment can detect events over a large range of recoil-electron energies, effectively measuring it with excellent precision. There is the possibility of filling DARWIN with liquid xenon depleted of the double-beta-decaying 136 Xe isotope. This would allow the study of higher energy solar neutrinos. Here we consider these two different scenarios, i.e., the 136 Xe-depleted version of DARWIN and the one where the abundance of 136 Xe agrees with natural expectations.
Other than the background from 136 Xe, for pp-neutrinos, the double electron capture decay of 124 Xe leads to two narrow peaks at 37 keV and 10 keV [1] and, at higher recoil energies, radioactive backgrounds from the detector components and the liquid volume supersede the pp-neutrino events for recoil kinetic energies above 200 keV or so. When simulating DARWIN data, we restrict our sample to events with recoil kinetic energies below 220 keV and assume that, in this energy range, the only sources of background are those from 136 Xe and 124 Xe. We simulate the backgrounds using the results published in [1]. When analyzing the simulated data, we marginalize over the normalization of the two 124 Xe lines, which we treat as free parameters, and the normalization of the 136 Xe recoil spectrum, which we assume is independently measured with 0.1% precision. We assume the shape of the 136 Xe recoil spectrum is known with infinite precision. For the 136 Xe-depleted version of DARWIN, we assume the 136 Xebackground is 1% of the background presented in [1]. We organize the simulated data into recoil-kinetic-energy bins with 10 keV width, consistent with the recoil-kinetic-energy resolution quoted in [1], starting at 1 keV. We use a simple χ 2 -test in order to address questions associated to the sensitivity of DARWIN to different parameters and in order to combine simulated DARWIN data with those from other experiments.

III. TESTING THE THREE-MASSIVE-NEUTRINOS PARADIGM
In the absence of more new physics, existing data reveals that the neutrino weak-interaction-eigenstates ν α , α = e, µ, τ , are linear combinations of the neutrino mass-eigenstates ν i with mass m i , i = 1, 2, 3: where the U αi , α = e, µ, τ , i = 1, 2, 3, define the elements of a unitary matrix. Here, we are only interested in solar neutrinos so all accessible observables are sensitive to |U ei | 2 , i = 1, 2, 3. These, in turn, are parameterized with two mixing angles, θ 12 and θ 13 . Following the parameterization of the Particle Data Group [14], and unitarity uniquely determines the third matrix-element-squared: Combined fits to the existing data reveal that the two independent mass-squared differences are ∆m 2 21 ≡ m 2 2 − m 2 1 ∼ 10 −4 eV 2 and |∆m 2 31 | ≡ m 2 3 − m 2 1 ∼ 10 −3 eV 2 . For more precise values see, for example, [15]. * While ∆m 2 21 is defined to be positive, the sign of ∆m 2 31 is still unknown; for our purposes here, it turns out, this is irrelevant. The two mixing parameters of interest have been measured quite precisely. According to [15], at the one-sigma level, The experiments that contribute most to these two measurements are qualitatively different. θ 12 is best constrained by solar neutrino experiments -and is often referred to as the "solar angle" -while θ 13 is best constrained by reactor antineutrino experiments -and is often referred to as the "reactor angle." We are interested in the solar pp-neutrinos. These have a continuous energy spectrum that peaks around 300 keV and terminates at around 420 keV. The matter-potential V = √ 2G F N e , where G F is the Fermi constant and N e is the electron number-density, inside the Sun is V < 2 × 10 −5 (eV 2 /MeV) so, for neutrino energies E < 0.420 MeV, given what is known about the neutrino mass-squared differences, matter effects can be neglected. Including the fact that, for all practical purposes, solar neutrinos lose flavor coherence as they find their way from the Sun to the Earth, it is trivial to show that the ν e survival probability is energy independent and given by On the other hand, solar neutrino experiments cannot distinguish ν µ from ν τ -the neutrino energies are too smallbut are potentially sensitive to the combination P ea ≡ P eµ + P eτ . In the three-massive-neutrinos paradigm Given our current knowledge of mixing parameters, for pp-neutrinos, we can indirectly infer that P ee = 0.552 ± 0.025, naively combining the uncertainties in Eq. (III.3) in quadrature. According to [1], after 20 ton-years of exposure, DARWIN can measure P ee with better than 1% accuracy. Assuming the three-massive-neutrinos paradigm, this can be converted into a measurement of the relevant mixing parameters. Fig. 1(top,left) depicts the allowed region of the sin 2 θ 12 × sin 2 θ 13 parameter space assuming DARWIN can measure P ee for pp-neutrinos at the 1% level, and assuming the best-fit value is P ee = 0.552. There is very strong degeneracy between different values of sin 2 θ 12 and sin 2 θ 13 , for obvious reasons. The degeneracies can be lifted by including constraints from other neutrino experiments.
It is interesting to investigate how well one can constrain neutrino-mixing parameters using only solar-neutrino data. In order to estimate that, we add to the hypothetical pp-neutrino measurement from DARWIN current information from 8 B neutrinos, mostly from the Super-Kamiokande and SNO experiments, see [16,17] and references therein. These provide the strongest constraints on sin 2 θ 12 . Here, we address this in a simplified but accurate way [18], postulating that 8 B experiments measure (III.6) * See also http://www.nu-fit.org. † For example, at the center of the Sun, for neutrino energies less than 420 keV, the "matter equivalent" of sin 2 2θ 12 differs from its vacuum counterpart by less than one percent. with 4% accuracy, consistent with the current uncertainty on sin 2 θ 12 , mostly constrained by high-energy solar neutrino data.
is the average probability that a 8 B neutrino arrives at the surface of the Earth as a ν 2 (or a ν 1 ). When the 8 B data is treated as outlined above, it translates into the open regions bound by dashed lines in Fig. 1(top,left). Strong matter effects lead to the boomerang-shaped allowed region of the parameter space and restrict the parameter space to values of sin 2 θ 12 0.5. The results of the joint pp− 8 B analysis are depicted in Fig. 1(top,right). All degeneracies present in the pp-neutrino data are lifted and one is constrained to small values of sin 2 θ 13 and sin 2 θ 12 < 0.5.
The combined 8 B and DARWIN data can rule out sin 2 θ 13 = 0 with some precision. This is important; it implies that a hypothetical DARWIN measurement of the pp-neutrino flux, combined with the current 8 B solar neutrino data, can measure sin 2 θ 13 in a way that is independent from all non-solar measurements. The marginalized χ 2 as a function of sin 2 θ 13 is depicted in Fig. 1(bottom), for 300 ton-years of simulated DARWIN data and the current 8 B sollar neutrino data. On average, if the pp-neutrino flux can be measured at the percent level, we expect to measure sin 2 θ 13 at the 35% level and rule out sin 2 θ 13 = 0 at almost the three-sigma level. Here we consider the two scenarios outlined earlier, one with natural xenon (dashed line), the other with 136 Xe-depleted xenon (solid line).
The precision on sin 2 θ 13 obtained above is not comparable to that of the current measurement of sin 2 θ 13 , Eq. (III.3). However, these measurements are qualitatively different. The most precise measurements of sin 2 θ 13 come from reactor antineutrino experiments and a baseline of order 1 km [19][20][21]. The estimate discussed above is a "solar only" measurement, i.e., it exclusively makes use of measurements of neutrinos (and not antineutrinos) produced in the Sun. Current measurements of sin 2 θ 13 that make use of neutrinos (as opposed to antineutrinos), from T2K and NOvA, are much less precise (at the 50%, see [22,23]). Looking further into the future, the DUNE experiment, for example, is expected to independently measure the "neutrino-only" value of sin 2 θ 13 at the 20% level [24] (or worse, depending on the assumptions made in the analysis).

IV. BEYOND THE THREE-MASSIVE-NEUTRINOS PARADIGM
The fact that the pp-neutrino flux can be computed with great precision, combined with the sub-MeV pp-neutrino energies, allows a high-statistics measurement of the pp-neutrino flux to meaningfully search for phenomena beyond the thee-massive-neutrinos paradigm. Here we concentrate on testing the hypothesis that the neutrinos produced in the Sun have a nonzero probability of behaving as "sterile neutrinos" ν s , characterized by their lack of participation in charged-current and neutral-current weak interactions.
We first discuss, in Sec. IV A, the case where the oscillation probabilities are energy-independent for the energies of interest, as in the case of the thee-massive-neutrinos paradigm discussed in Sec. III. In particular, we test the hypothesis that P ee + P ea = 1 for pp-neutrinos. Then, in Sec. IV B, we compute DARWIN's ability to constrain the hypothesis that there is a fourth neutrino ν 4 and that its mass is quasi-degenerate with m 1 .

A. Model-Independent Considerations
As discussed in Sec. II, we are interested in the shape and normalization of the electron recoil-energy spectrum from neutrino-electron elastic scattering. The differential cross-section for ν e and ν a scattering are different, both in normalization and shape and hence, in principle, one can obtain independent information on both P ee and P ea .
We simulate and analyze 300 ton-years of DARWIN pp-data, as discussed in Sec. II, and attempt to measure P ee and P ea independently. The results are depicted in Fig. 2(left) for both the natural xenon (dashed) and the 136 Xe-depleted (solid) hypotheses. Strong departures from P ee + P ea = 1 are allowed and the "natural" data are not capable of ruling out P ea = 0 at the three-sigma confidence level. The "depleted" data can rule out P ea = 0 at the five-sigma confidence level. For both scenarios, one can constrain the departure of P ee + P ea from one, which we interpret as the oscillation probability into sterile neutrinos P es ≡ 1 − P ee − P ea . The colorful diagonal lines in Fig. 2(left) correspond to different constant values of P es . Fig. 2(right) depicts χ 2 as a function of P es , marginalized over P ee and restricing P es to non-negative values for both scenarios. If DARWIN data are consistent with the three-active-neutrinos paradigm, they will be capable of constraining P es < 0.35 at the two-sigma confidence level even if DARWIN is filled with natural xenon.

B. Fourth-Neutrino Hypothesis
We explore in more detail the scenario where there is one extra neutrino mass-eigenstate ν 4 with mass m 4 . In this case, the interaction eigenstates are, including ν s , related to the four mass-eigenstates via a 4 × 4 unitary matrix U αi , α = e, µ, τ, s, i = 1, 2, 3, 4. We will concentrate on the scenario where, among the four U si , only U s1 and U s4 are potentially nonzero. ‡ In this case, we can parameterize the |U ei | 2 entries of the mixing matrix using three mixing ‡ It is easy to generalize this analysis assuming that only one of the U sj , j = 1, 2, 3, and U s4 are potentially nonzero. We spell this out in the Appendix. Given the quasi-two-flavors nature of these solar neutrino oscillations, to be discussed momentarily, the entire physical parameter space is spanned by either fixing ∆m 2 41 > 0 and allowing sin 2 θ 14 ∈ [0, 1] or allowing both signs for ∆m 2 41 and restricting sin 2 θ 14 ∈ [0, 0.5] in such a way that ν 4 is always "mostly sterile." Here, the former convention -to fix the sign of ∆m 2 41 > 0 -is most convenient. With this choice, when sin 2 θ 14 ∈ [0, 0.5], the heaviest of the two quasi-degenerate states (i.e., ν 4 ) is mostly sterile, when sin 2 θ 14 ∈ [0.5, 1], the lightest among the two quasi-degenerate states (i.e., ν 1 ) is mostly sterile. For historical reasons, we will refer to sin 2 θ 14 ∈ [0, 0.5] as the light side of the parameter space and sin 2 θ 14 ∈ [0.5, 1] as the dark side [25].
We are interested in the hypothesis that ∆m 2 41 ∆m 2 21 and outside the reach of all current neutrino experiments. In this case, the current neutrino oscillation data constrain the oscillation parameters ∆m 2 21 , ∆m 2 31 , sin 2 θ 12 , and sin 2 θ 13 exactly as in the three-massive-neutrinos paradigm. Furthermore, builiding on the discussion in Sec. III, it is easy to conclude that the oscillation probabilities of interest P eα , α = e, a, s, are only functions of sin 2 θ 12 , sin 2 θ 13 , sin 2 θ 14 , and ∆m 2 41 . Further taking advantage of the fact that |∆m 2 21 |/(2E), |∆m 2 31 |/(2E) V , it is straightforward to compute where P 2f ee is the survival probability obtained in the scenario where there are only two flavors, ν 2f e and ν 2f s , characterized by the mass-squared difference ∆m 2 41 and the mixing angle θ 14 , defined via ν 2f e = cos θ 14 ν 1 + sin θ 14 ν 4 . Inside P 2f ee , the matter potential is replaced by an effective matter potential V eff . It takes into account the neutral-current contribution to the matter potential V N C = − √ 2/2G F N n , where N n is the neutron number density in the medium, while the charged-current contribution is rescaled by (1 − |U 2 e2 − |U e3 | 2 ) = cos 2 θ 13 cos 2 θ 12 : In the sun, the position-dependency of the electron and neutron number densities are slightly different [2]. § In the sun's core, N n is around 50% of N e and V eff is slightly less than one half of the standard matter potential in the Sun. Eq. (IV.3) allows us to estimate, in very general terms, the impact of the sterile neutrinos. For P 2f ee = 1, we recover the three-active-neutrinos result, P ee = cos 4 θ 12 cos 4 θ 13 + sin 4 θ 12 cos 4 θ 13 + sin 4 θ 13 , Eq. (III.4). On the other hand, for P 2f ee = 0, P ee = sin 4 θ 12 cos 4 θ 13 + sin 4 θ 13 such that, given the current knowledge of oscillation parameters, P ee ∈ [0.09, 0.55]. (IV.7) P es values, on the other hand, are allowed to be as small as zero and as large as 0.68. In Sec. IV A, we discussed that, very generically, DARWIN can rule out P es < 0.35 at the two-sigma level. The situation here is more constrained as P ee , P ea , P es are not only required to add up to one but depend on the same oscillation parameters. We proceed to discuss the sensitivity of DARWIN to the new oscillation parameters ∆m 2 41 , sin 2 θ 14 by taking advantage of the fact that the properties of P 2f ee are well known (see, for example, [26]). Here, it is impossible to distinguish the light from the dark side of the parameter space since the oscillation probabilities are invariant under sin 2 θ 14 ↔ 1−sin 2 θ 14 . Varying sin 2 2θ 14 ∈ [0, 1], Eqs. (IV.9) define a line segment in the P ee ×P eaplane, depicted in Fig. 2(left) -burgundy line with positive slope -keeping in mind the segment extends to P ee values below 0.4. Fig. 3 depicts χ 2 as a function of sin 2 θ 14 in the regime where Eqs. (IV.9) are a good approximation, for 300 ton-years of simulated DARWIN data, for both the natural (dashed) and 136 Xe-depleted (solid) scenarios. In this analysis, and in the upcoming analyses discussed this subsection, we assume that sin 2 θ 13 and sin 2 θ 12 are known with infinite precision. This is, currently, a good approximation for sin 2 θ 13 and will be a good approximation for sin 2 θ 12 once data from the JUNO experiment is analyzed [27]. Similar results were recently presented and discussed in [28]. Where are assumptions agree, the estimated sensitivity also agrees.
For intermediate values of ∆m 2 41 , P 2f ee is well describe by the strong MSW effect in the adiabatic regime. In this case, for a range of energies, Here, oscillation probabilities are very different in the light and dark sides. In particular, in the light side of the parameter space P ee (P es ) is small (large) and increases (decreases) linearly with sin 2 θ 14 . If DARWIN data are consistent with three-active neutrinos, in this region of parameter space, small values of sin 2 θ 14 will be excluded § There are relatively more neutrons in the center of the sun relative to its edges. This is due to the fact that most of the solar helium is concentrated in the core. 10 −11 eV 2 . This means that, for 10 −11 ∆m 2 41 /(eV 2 ) 10 −9 , the oscillation probabilities are well described by Eqs. (IV.9). Fig. 5 depicts contours of constant P ee in the ∆m 2 41 × sin 2 θ 14 -plane for E ν = 300 keV. The other parameters are fixed to their current best-fit values, Eq. (III.3). We assume the matter potential is spherically symmetric and drops exponentially, V eff ∝ e −r/r s 0 . We fit information from the prediction of the B16-GS98 solar model [4] and obtain r s 0 = R /10.37 where R = 6.96 × 10 11 m is the average radius of the Sun; see Fig. 4 for a comparison of the matter potential in the standard case (left) and in the scenario of interest here (right, labeled sterile neutrino).
Under these circumstances, P 2f ee can be computed exactly [29]. For simplified pedagogical discussions see, for example, [26,30]. We assume all solar neutrinos are produced in the exact center of the Sun; we explicitly verified that the results we get are very similar to the results we would have obtained by integrating over the region where pp-neutrinos are produced. The region where matter effects are strong and the adiabatic condition holds correspond to the vertical sides of the constant P ee regions that form quasi-triangles. The "return" to vacuum oscillations at low and high values of ∆m 2 41 is highlighted by the vertical, dark lines, which correspond to constant values of the averaged-out vacuum oscillation probability. For a detailed discussion of the boundary between the adiabatic and non-adiabatic transition, including L dependent effects, see [31]. We simulate 300 ton-years of DARWIN data consistent with the three-massive-neutrinos paradigm and assuming the true values of sin 2 θ 12 and sin 2 θ 13 are the ones in Eq. (III.3). We restrict our discussion to values of ∆m 2 41 < 10 −6 eV 2 . Larger values are constrained by measurements of higher-energy solar neutrinos; these constraints have been explored in [32,33], along with a detailed discussion of the oscillation probabilities. The expressions we derive here are contained in the analyses of [32,33] if one explores them in the appropriate regime. Fig. 6 depicts the region of the tan 2 θ × ∆m 2 41 -plane inside of which 300 ton-years of DARWIN data is sensitive, at the 90% confidence level, to the fourth neutrino for both the natural-xenon scenario (dashed line) and the depleted-136 Xe scenario (solid). On a log-scale, the contour is symmetric relative to tan 2 θ 14 = 1 when one cannot distinguish the light from the dark side of the parameter space [25], a feature one readily observes, as advertised, for small values of ∆m 2 41 . The impact of nontrivial matter effects is also readily observable. For larger values of the ∆m 2 41 , the sensitivity to small mixing angles is expected to "shut-off" quickly -see The low energies of the pp-neutrinos combined with the long Earth-Sun distance render DARWIN a specially powerful probe of the hypothesis that neutrinos are pseudo-Dirac fermions. This is the hypothesis that there are right-handed neutrinos coupled to the left-handed lepton doublets and the Higgs doublet via a tiny Yukawa coupling y and that lepton number is only slightly violated. In these scenarios, each of the neutrino mass eigenstates is "split" into two quasi-degenerate Majorana fermions, each a 50-50 mixture of an active neutrino (from the lepton doublet) and a sterile neutrino (the right-handed neutrino). The mass splitting is small enough that, for most applications, the two quasi-degenerate state act as one Dirac fermion. Pseudo-Dirac neutrinos reveal themselves via active-sterile oscillations associated with very large mixing and very small mass-squared differences.
In the language introduced here, a pseudo-Dirac neutrino corresponds to sin 2 2θ 14 = 1 (maximal mixing) and the small mass splitting leads to a nonzero ∆m 2 41 = 4 m 1 where m 1 ± are the masses of the two quasi-degenerate states (here ν 1 and ν 4 ), m 1 is the Dirac mass, proportional to the neutrino Yukawa coupling and characterizes the strength of the lepton-number violating physics. Fig. 7 depicts χ 2 as a function of ∆m 2 41 for sin 2 2θ 14 = 1 associated with 300 ton-years of simulated DARWIN data for both the natural xenon (dashed) and the 136 Xe-depleted (solid) scenarios, assuming the data are consistent with no new neutrino states. Current solar neutrino data exclude ∆m 2 41 values larger than 10 −12 eV 2 [9,10] so DARWIN can extend the sensitivity to ∆m 2 41 -and -by an order of magnitude.

V. CONCLUDING REMARKS
Next-generation WIMP-dark-matter-search experiments will be exposed to a large-enough flux of solar neutrinos that neutrino-mediated events are unavoidable. The DARWIN collaboration recently argued that DARWIN can collect a large, useful sample of solar pp-neutrinos, detected via elastic neutrino-electron scattering [1], and measure the survival probability of pp-neutrinos with sub-percent precision. Here we explored the physics potential of such a sample in more detail, addressing other concrete neutrino-physics questions and exploring whether one can also extract information from a precise measurement of the shape of the differential pp-neutrino flux. We estimate that, with 300 ton-years of data, DARWIN can not only measure the survival probability of pp-neutrinos with sub-percent precision but also determine, with the help of current solar neutrino data, the value of sin 2 θ 13 , with the potential to exclude sin 2 θ 13 = 0 close to the three sigma level. Such a pp-neutrino sample would allow one to perform a "neutrinos-only" (and solar-neutrinos-only) measurement of sin 2 θ 13 and sin 2 θ 12 . Such a measurement can be compared with, for example, reactor-based "antineutrinos-only" measurements of the same mixing parameters and allow for nontrivial tests of the CPT-theorem and other new physics scenarios.
DARWIN can also test the hypothesis that pp-neutrinos are oscillating into a combination of active and sterile neutrinos. We estimate that DARWIN data can exclude the hypothesis that the pp-neutrinos are "disappearing" in an energy independent way -assuming their data are consistent with the three-active-neutrinos paradigm -especially if the experiment manages to fill their detector with 136 Xe-depleted xenon.
We explored in some detail how well DARWIN can constrain the existence of a new neutrino mass-eigenstate ν 4 (mass m 4 ) that is quasi-degenerate and mixes with ν 1 , i.e, ∆m 2 41 ∆m 2 21 , U s1 , U s4 = 0, U s2 = U s3 = 0. Our estimated sensitivity is depicted in Fig. 6. It supersedes that of all current and near-future searches for new, very light neutrinos. In particular, DARWIN can test the hypothesis that ν 1 is a pseudo-Dirac fermion as long as the induced mass-squared difference is larger than 10 −13 eV 2 . This is one order of magnitude more sensitive than existing constraints [9,10].