Obsidian/Zotero/001_artiklid/shekarsearchingbeyondstandardmodel2025.md

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category: literaturenote citekey: shekarsearchingbeyondstandardmodel2025 title: Searching for beyond-Standard-Model solar neutrino interactions using directional detectors authors: "Shekar, Anirudh Chandra; Lisotti, Chiara; Mishra, Nityasa; O'Hare, Ciaran A. J.; Strigari, Louis E." year: 2025 date: 2025-00-00 2025 url: "https://arxiv.org/abs/2506.09322v1" zotero_key: D86TPPVG zotero_storage: V2GQF3XD collections: transpordi_planeerimine folder: 001_artiklid firstAuthor: "Shekar, Anirudh Chandra"

status: converted

Searching for beyond-Standard-Model solar neutrino interactions using directional detectors

Anirudh Chandra Shekar,1, a Chiara Lisotti,2, b Nityasa Mishra,1, c Ciaran A. J. O'Hare,2, d and Louis E. Strigari1, e

¹Department of Physics and Astronomy, Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, Texas 77843, USA

²ARC Centre of Excellence for Dark Matter Particle Physics, The University of Sydney, School of Physics, NSW 2006, Australia

Micro-pattern gaseous detectors (MPGDs) are a class of technologies that enable the full threedimensional spatial reconstruction of ionisation tracks from nuclear and electron recoils in gas. Anticipating near-future 30 m³ -scale time projection chambers with MPGD-based readout, we forecast the sensitivity of such directionally-sensitive low-energy recoil detectors to neutrino interactions beyond the Standard Model. We work in the framework of neutrino non-standard interactions (NSIs), and calculate the combined recoil energy-angle distributions of the electron recoil signal generated by solar neutrinos in atmospheric-pressure He:CF⁴ gas. We estimate the expected exclusion limits that such an experiment could place on various NSI parameters, as well as the mass and coupling of a new light mediator that interacts with electrons and neutrinos. We find that with an achievable background reduction of around a factor of ten from current estimates for a 30 m³ optical-readout detector using this gas mixture, sensitivity to NSI parameters would already approach Borexino's sensitivity. Directionality also allows for event-by-event neutrino energy reconstruction, which would provide a means to resolve some parameter degeneracies present in the modified neutrino cross section in this formalism. Our results strongly motivate the development of small-scale directionally-sensitive gas detectors for neutrino physics.

I. INTRODUCTION

Almost seventy years since their first experimental detection, the fundamental nature of the neutrino and its connection to the rest of the otherwise well-tested Standard Model (SM) of particle physics is still mysterious. Although neutrinos remain challenging to detect, the pursuit of answers to questions such as the origin of their mass and the precise phenomenology of their oscillations has accelerated the proliferation of many diverse classes of large and highly sensitive detectors, see e.g. Refs. [1– 21] for an incomplete sample. Detectors are able to observe and study neutrinos from a range of sources at the precision level, extending from low-energy neutrinos created by human beings, to ultra-high-energy neutrinos generated by extreme astrophysical environments, see e.g. Ref. [22] for a review.

Although most successive generations of neutrino observatories have increased in size over the years, technological developments have also allowed for the size of a detector that is capable of making competitive measurements of a neutrino source to decrease—for example by trading target mass for lower energy thresholds, better signal reconstruction, and enhanced background rejection. In the case of neutrinos produced by the Sun, the first hints of detection by the liquid-xenon-based XENONnT [23] and PandaX [24] experiments are setting new records for the smallest physical size of a detector able to observe neutrinos from an astrophysical source.

These recent detections are possible thanks to the combination of the large cross section for coherent elastic neutrino-nucleus scattering (CEνNS) [25, 26] as well as the excellent nuclear/electron recoil discrimination and low backgrounds of modern dual-phase xenon time projection chambers—the latter advance having primarily been driven by the (so-far unsuccessful) hunt for galactic dark matter particles interacting with nuclei [27– 29]. These results, when combined with the experimental measurements of CEνNS by COHERENT [30–32] and CONUS+ [33], are inspiring explorations into the possible manifestations of beyond-SM neutrino interactions at lower energies than have been studied before, see e.g. Refs. [34–54]. However, at the ≳keV recoil energy scale searched for by these low-background detectors, the CEνNS process is actually responsible for a subdominant number of neutrino events. In terms of total numbers, elastic neutrino-electron scattering (EνeS) due to the solar pp flux is a stronger signal by around an order of magnitude. Unfortunately, at present, these events are hidden beneath a large background of electron recoils due to radioactivity in the detector and environment, making it challenging to perform precision measurements of the neutrino-electron cross section. In the long-term, future xenon experiments like XLZD [55–58] should be poised to set constraints on new physics manifesting in low-energy EνeS at an interesting level of sensitivity (given estimates for the sensitivity of the current generation like LZ, XENONnT and PandaX [34, 46, 59– 62]), but this will depend strongly on the level of electron recoil background that is achieved. At present, the best limits still come from Borexino [63–70], whose sensitivity

^a anirudhcs@tamu.edu

^b maria.lisotti@sydney.edu.au

^c nityasa mishra@tamu.edu

^d ciaran.ohare@sydney.edu.au

e strigari@tamu.edu

mostly (but not entirely) supersedes [71] that of experiments using terrestrial neutrino sources like beam-dumps or reactors [72–76].

Nevertheless, due to the relatively high energy threshold of Borexino when compared to dark matter detectors, attempting to test beyond-SM physics in the neutrino sector using low-energy electron recoils remains a well-motivated challenge. To meet this challenge, however, we require detectors that are able to accumulate a sufficient number of reliable neutrino events that are not buried beneath backgrounds which would otherwise preclude the robust measurements of their cross section required to discern differences from the SM prediction. A promising idea to achieve this—one which was proposed many years ago in this context [77–79] , but has only become experimentally feasible recently—is to attempt the full spatial reconstruction of the recoil track directions.

Although neutrino directionality is an important and well-established observable at higher energies [4, 9, 18, 81], there are currently only a very limited number of techniques able to reconstruct this information in the regime relevant for the low-energy solar neutrino fluxes [82–85]. So far, directional detection of lowenergy recoil tracks at the necessary O(10–100) keV energies has only been shown to be experimentally feasible for time-projection chambers (TPCs) using gaseous targets [86, 87]. This technology has seen significant advances over the last few decades, driven partially by the goal of finding a dark matter detector that is able to circumvent [88–92] the so-called the "neutrino fog" limit [28, 93–100] imposed upon conventional direct detection experiments. It has been shown that a costbalanced trade-off is possible using the high-resolution capabilities of highly-segmented micro-pattern gas detectors (MPGDs), which enable sub-mm spatial resolution and high signal-to-noise reconstructions of fully 3 dimensional ionisation tracks [87, 101–103].2

The reconstruction of ionisation tracks in gas can provide a large amount of event-level information about the recoils that generated them. However, the detrimental effects of diffusion require that the distance over which this ionisation is drifted through the gas to the readout plane must be kept to a minimum, i.e. 25–50 cm at most. This in turn implies that a large-scale experiment of, say, 100 m³ volume or more, could only be achieved in a modular configuration composed of multiple 10 m³ -scale detectors. Working towards the ultimate vision of a large-scale "recoil observatory" of this kind is the goal of both the Cygnus Consortium [101, 111–114] and the CYGNO collaboration [115–119]. The scale of

an experiment necessary for a dark matter search into the neutrino fog would be naturally preceded by several stepping-stone detectors of ^O(10 m³ ) volume. These experiments— which are in development now [119]—are the focus of this study as they will already have a sufficient target mass and background level to obtain sensitivity to solar neutrinos via EνeS.

As for the science case supporting a gas detector of this scale over, say, a non-directional liquid-based detector, there are several notable advantages a directional measurement offers. In particular, monodirectional sources like the Sun or a neutrino beam are interesting because of the fixed kinematic relationship between the neutrino's initial energy and the energy and scattering angle of the subsequent recoil. In principle, a simultaneous measurement of the recoil energy and angle (as enabled by MPGDs) allows for the event-by-event reconstruction of the neutrino energy spectrum. On top of this, a detector able to reconstruct the full spatial distributions of recoil-induced ionisation clouds comes with a powerful intrinsic method for background rejection and nuclear/electron recoil discrimination, both of which have been shown to work down to ∼10 keV electron recoil energies [113, 115, 120–126].

All of the advantages of directional gas detectors mentioned above inspired Ref. [127], which evaluated the detector requirements to perform solar neutrino flux spectroscopy using EνeS. Here, we consider the extent to which the science case for this kind of measurement extends beyond this. As well as simply reconstructing the neutrino flux, there will be additional, more subtle information about the neutrino-electron cross section accessible to an experiment measuring angular information that might uncover corrections originating from new physics. So in this paper, we extend Ref. [127] by evaluating the size and performance of a gas-based directional recoil detector required to test whether or not neutrino-electron scattering kinematics continues to be described by the SM at low energies.

The paper is structured as follows. In Sec. II we describe the theoretical calculation of the directiondependent scattering rate of solar neutrinos with electrons, within and beyond the SM. In Sec. III we describe how our model for the detector technology and discuss the relevant effects that limit its ability to reconstruct the energy and direction of neutrino events. In Sec. IV we explain our statistical methodology, and then present results in Sec. V.

II. NEUTRINO-ELECTRON SCATTERING

We begin by describing neutrino-electron elastic scattering, both within the SM and after allowing for the possibility of so-called "Non-Standard Interactions" (NSIs) due to the presence of new mediators beyond the SM. We also describe the helpful density matrix formalism, which we adopt to handle the effects of NSIs on neutrino

¹ See also Ref. [80] for a similar proposal in the context of geoneutrinos.

² This type of directional detection is referred to as "recoil imaging" [87, 102] to distinguish it from other, less powerful types of directional detection that seek to only measure spatial projections of ionisation [104, 105] or to infer directionality indirectly [106–110], e.g. via daily modulation signals.

scattering and oscillations in this formalism.

A. Directional event rate

For a detector containing N^T scattering targets (e.g. electrons or nuclei) per unit detector mass, the differential event rate for elastic neutrino scattering on those targets is given by,

$$\frac{\mathrm{d}R}{\mathrm{d}E_r} = NT \int{E\nu^{\rm min}}^{E\nu^{\rm max}} \mathrm{d}E\nu \frac{\mathrm{d}\Phi}{\mathrm{d}E\nu} \sum\alpha P{e\alpha}(E\nu) \frac{\mathrm{d}\sigma\alpha}{\mathrm{d}E_r} \,. \tag{1}$$

Here dΦ/dE^ν is the neutrino flux, E^r is the recoil energy, and E^ν is the neutrino energy. The quantity dσα/dE^r is the differential scattering cross section for the interaction of an α-flavoured neutrino with the target particle. The neutrino-energy-dependent function Peα(Eν) is the probability that the electron neutrino ν^e produced in the solar core is detected as a neutrino with flavour α = {e, µ, τ} after oscillations. It is not possible to identify any flavour information at the level of individual recoil events, so we sum over α when computing dR/dEr.

To describe the directional dependence of the recoil event rate, we follow the discussion in Refs. [89, 128]. For a detector with directional sensitivity, we expand the space of observables to include the recoil direction ˆqr, in addition to the recoil energy, Er. To find the distribution of recoil energies and angles we compute the double-differential event rate, d²R/dErdΩr, where dΩ^r is a solid angle element around ˆqr. The expression for this is the following,

$$\frac{\mathrm{d}^{2}R}{\mathrm{d}E{r}\mathrm{d}\Omega{r}} = N{T} \int \mathrm{d}E{\nu}\mathrm{d}\Omega{\nu} \sum{\alpha} P{e\alpha}(E{\nu}) \frac{\mathrm{d}^{2}\Phi}{\mathrm{d}E{\nu}\mathrm{d}\Omega{\nu}} \frac{\mathrm{d}^{2}\sigma{\alpha}}{\mathrm{d}E{r}\mathrm{d}\Omega_{r}}.$$ (2)

The first quantity inside the integral is the doubledifferential neutrino flux, which in the case of the Sun can be written straightforwardly as,

$$\frac{\mathrm{d}^2 \Phi}{\mathrm{d} E{\nu} \mathrm{d} \Omega{\nu}} = \frac{\mathrm{d} \Phi}{\mathrm{d} E{\nu}} \delta \left( \hat{q}{\nu} - \hat{q}_{\odot} \right) , \qquad (3)$$

where ˆq^ν is the incoming neutrino direction and ˆq^⊙ is the unit vector pointing from the Sun to the detector. As for the double-differential cross section, we can assume the scattering is azimuthally symmetric, which means our solid angle element is dΩ^r = 2πd cos θr, or dΩ^r = 2πd cos θ^⊙ if we work in a heliocentric coordinate system. The scattering kinematics relates this angle to the neutrino energy and the energy of the recoil via,

$$\cos \thetar = \hat{q}{\nu} \cdot \hat{q}r = \frac{E{\nu} + m}{E_{\nu}} \sqrt{\frac{E_r}{E_r + 2m}}, \qquad (4)$$

so that

$$\frac{\mathrm{d}^2 \sigma_{\alpha}}{\mathrm{d} E_r \mathrm{d} \Omegar} = \frac{1}{2\pi} \frac{\mathrm{d} \sigma{\alpha}}{\mathrm{d} Er} \delta \left( \hat{q}{\nu} \cdot \hat{q}r - \frac{E{\nu} + m}{E_{\nu}} \sqrt{\frac{E_r}{E_r + 2m}} \right), \tag{5}$$

where m is the mass of the target particle. Using Eqs.(3) and (5) in (2), we find,

$$\frac{\mathrm{d}^{2}R}{\mathrm{d}E{r}\mathrm{d}\Omega{r}} = \frac{N{T}}{2\pi} \int \mathrm{d}E{\nu} \sum{\alpha} P{e\alpha}(E{\nu}) \frac{\mathrm{d}\sigma{\alpha}}{\mathrm{d}E{r}} \frac{\mathrm{d}\Phi}{\mathrm{d}E{\nu}} \times$$

$$\delta\left(\hat{q}{r} \cdot \hat{q}{\odot} - \frac{E{\nu} + m}{E{\nu}} \sqrt{\frac{E{r}}{E{r} + 2m}}\right). \tag{6}$$

The δ-function in Eq.(6) can be rewritten as δ x + 1 E where we define:

$$x = -\frac{1}{E_{\nu}}, \quad y = \sqrt{\frac{m^2 E_r}{E_r + 2m}}, \quad \frac{1}{\mathcal{E}} = \frac{\hat{q}r \cdot \hat{q}{\odot}}{y} - \frac{1}{m}, \quad (7)$$

where ˆq^r · qˆ^⊙ = cos θ⊙. We can then perform the δfunction integral over this new variable x to get

$$\frac{\mathrm{d}^2 R}{\mathrm{d} E_r \mathrm{d} \Omega_r} = \frac{NT}{2\pi} \frac{\mathcal{E}^2}{y} \left. \left( \sum{\alpha} P{e\alpha} \frac{\mathrm{d} \sigma{\alpha}}{\mathrm{d} Er} \frac{\mathrm{d} \Phi}{\mathrm{d} E{\nu}} \right) \right|{E{\nu} = \mathcal{E}} . \tag{8}$$

1. Oscillation probabilities

The Sun produces electron-flavoured neutrinos ν^e at its core via nuclear processes [129]. The neutrino flavour states then evolve during propagation through the solar matter and in the vacuum until they are detected on Earth. The Hamiltonian for neutrino propagation from the core of the Sun to the Earth in the flavour basis is given by,

$$H = \frac{1}{2E{\nu}} U \begin{pmatrix} 0 & 0 & 0 \ 0 & \Delta m{21}^2 & 0 \ 0 & 0 & \Delta m_{31}^2 \end{pmatrix} U^{\dagger}$$

$$+ \sqrt{2} G_F N_e(x) \begin{pmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} ,$$

$$(9)$$

where U is the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix that transforms flavour eigenstates ν^α to mass eigenstates νⁱ , and the mass-squared differences that govern the mixing are labelled as ∆m² ij ^{≡ m2 i − m2} j . The term ^√ 2G^F Ne(x) is the matter potential, capturing the coherent forward scattering of neutrinos in the interior of the Sun. This term depends upon the number density of electrons in the Sun, Ne(x), which is a function of position x, which is measured relative to the centre of the Sun.

At the detector, the neutrino flavour state ν^α can be expressed as a superposition of νⁱ

$$P{e\alpha} = \sum{i=1}^{3} P^{m} (\nu{e} \to \nu{i}) |U_{\alpha i}|^{2},$$ (10)

where P ^m (ν^{e → ν}i) is the probability of adiabatic transition of ν^e to νⁱ in the Sun. For a more detailed discussion and derivation of Eq.(10), we refer to Refs. [130–132].

2. Standard Model neutrino cross section

The expression for d2R/dErdΩ^r derived above applies to both electron and nuclear scattering. However, we will now focus our attention towards EνeS for the remainder of this study. The motivation behind this is simply that the event rate from solar pp and ⁷Be neutrinos is within reach of the ^∼30 m³ -scale gas detectors currently under preparation. Nuclear recoils from the ⁸B neutrino flux could also be measured, but require much larger gas volumes (^∼ 1000 m³ ) as well as nuclear-recoil energy thresholds that still remain challenging (∼ keVr). We relegate a discussion of the prospects for measuring solar NSIs through CEνNS to Appendix B.

The SM differential cross section for EνeS (ναe ^{− →} ναe ⁻) is,

$$\frac{\mathrm{d}\sigma{\alpha}}{\mathrm{d}E{r}} = \frac{2G{F}^{2}m{e}}{\pi} \left[ g{L\alpha}^{2} + g{R}^{2} \left( 1 - \frac{E{r}}{E{\nu}} \right)^{2} - 2g{L\alpha}g{R} \frac{m{e}E{r}}{2E_{\nu}^{2}} \right] \tag{11}$$

where m^e is the electron mass, G^F is the Fermi constant, and g^L and g^R are the SM couplings. The process νee ^{− → ν}e^{e −} can occur through both a charged current via W exchange, as well as a neutral current via a Z. On the other hand the process νµ/τ e ^{− → ν}µ/τ ^{e −} is purely a neutral current interaction. As a result, the flavourdependent SM couplings are given by,

$$g_{Le} = 1 + \left(\sin^2 \theta_w - \frac{1}{2}\right),$$

$$g{L\mu} = g{L\tau} = \sin^2 \theta_w - \frac{1}{2},$$

$$g_R = \sin^2 \theta_w,$$ (12)

where sin² θ^w = 0.2387 is the weak mixing angle [133]. The differential cross section for ν^µ interactions with e is the same as ν^τ interactions at tree level, but there are percent-level differences between the two with higherloop corrections (see e.g. Ref. [131, 132]). Since an experiment of the scale we are interested in here will not be sensitive to these differences, we neglect any radiative corrections to Eq.(11).

B. NSI formalism

We have now laid out the expected rate of electron recoils due to solar neutrinos under the SM. That calculation will form our null hypothesis, which we will test against a possible extension to the SM. The extension we consider is expressed in terms of the popular nonstandard neutrino interactions (NSI) framework. For concreteness, we will consider NSIs due to a new vector mediator and examine two limiting regimes for its mass. A "light mediator" refers to one whose mass is comparable to, or smaller than, the typical momentum transfer q involved in the scattering process, while a "heavy mediator" is one whose mass is considerably higher than q. We expect qualitatively similar results for axial-vector, so for brevity, we do not explore these cases here (compare figures for these cases in e.g. Refs. [34, 46, 48–51, 71]). We emphasise that our motivation in this study is to determine whether the size, performance and background of a 30 m³ -scale directional detector is sufficient to distinguish NSIs from the SM at a competitive level—an exhaustive comparison of phenomenologically similar models is unnecessary to arrive at such a conclusion.

Let us assume the existence of a new vector mediator Z ′ of a gauged U(1) symmetry extending the SM. The mediator will couple to fermions f and neutrinos να. The NSI Lagrangian in this case can be written as [62, 71, 134, 135],

$$\mathcal{L}^{\text{NSI}} \subset g{Z'} \left( \sum{\alpha} Q{Z'}^{\nu{\alpha}} \bar{\nu}{\alpha L} \gamma^{\mu} \nu{\alpha L} + \sum{f} Q{Z'}^{f} \bar{f} \gamma^{\mu} f \right) Z'_{\mu}. \tag{13}$$

Here, gZ′ is the NSI coupling and Q^{i Z}′ denotes the individual model-dependent vector charges. A few specific examples of anomaly-free realisations of such models are given in Table I. In this study, we will show results for only the U(1)B−^L case, since the results for the various models are very similar. In the light mediator limit, the free parameters of the model are (gZ′ , mZ′ ), where mZ′ is the mediator mass.

Model	e/νe Q Z′	µ/νµ Q Z′	τ/ντ Q Z′	u/d Q Z′
B − L	−1	−1	−1	1/3
Le − Lµ	1	−1	0	0
Le − Lτ	1	0	−1	0

TABLE I. Q ^Z′ values for a few models

When the mass of the mediator significantly exceeds the momentum transfer of the interaction (mZ′ ≫ q), the Lagrangian of Eq.(13) can be approximated as an effective four-fermion operator interaction similar to the SM Lagrangian [71, 136–138],

$$\mathcal{L}^{\text{eff,NSI}} \subset -2\sqrt{2}GF \sum{f,P,\alpha,\beta} \varepsilon{\alpha\beta}^{f,P} \left(\bar{\nu}{\alpha}\gamma{\mu}L\nu{\beta}\right) \left(\bar{f}\gamma^{\mu}Pf\right),$$ (14)

where P = {L, R} is the projection operator. Here, we have introduced the dimensionless effective coupling strengths ε f,P αβ , which we take to be real-valued. For vector NSIs in the heavy mediator limit, we take the free parameters of the model to be the following combination of L and R couplings,

$$\varepsilon{\alpha\beta}^f \equiv \varepsilon{\alpha\beta}^{f,L} + \varepsilon_{\alpha\beta}^{f,R} \,. \tag{15}$$

In contrast to this, the axial-vector NSI coupling is ˜ε f αβ ≡ ε f,L αβ − ε f,R αβ . We will focus on the pure vector case in this work by setting the axial-vector coupling ˜ε αβ to zero, implying in turn that ε e,L αβ = ε e,R αβ .

We label flavours as α, β ∈ {e, µ, τ}. Cases where α = β signify "flavour-conserving" interactions while α ̸= β are "flavour-violating". The models quoted in Table I are all flavour-conserving, but it is possible to include flavour-violating terms in analogy with Eq.(14), see e.g. Ref. [139].

C. Density matrix formalism

We now move on to describe the density matrix formalism, which we adopt to handle the fact that the introduction of NSIs modifies both the neutrino oscillations and scattering at the same time. For more details on this formalism, see Refs. [46, 71, 140].

We first write down the labelling for the scattering processes, which will be,

$$\nu_{\alpha}T \to \nu_f T$$ ,

where T represents the scattering target, ν^α refers to the neutrino flavour state taking part in the scattering, and ν^f is the final neutrino flavour state. As mentioned above, NSIs open up the possibility of flavour-violating terms where f ̸= α.

In addition to scattering, neutrino oscillations allow the neutrino flavour at detection (να) to be different from its flavour at production—ν^e for the case of solar neutrinos—which we label as,

$$\nue \to \nu\alpha$$ .

In order to account for the combined effect of neutrino propagation and neutrino detection, one needs to combine the two processes above, summing over the intermediate state να.

Let Sint and Sprop be the S-matrices for scattering and oscillation, respectively. The square of the amplitude of the combined process can be written as,

$$|A_{\nu_e \to \nuf}|^2 = \left| \sum{\alpha} \langle \nuf | S{\text{int}} | \nu\alpha \rangle \langle \nu\alpha | S_{\text{prop}} | \nu_e \rangle \right|^2.$$ (16)

Any individual detected recoil event could be associated with any neutrino flavour. So we take the sum over amplitudes for all f when computing the rate,

$$\frac{\mathrm{d}R}{\mathrm{d}E_r} \propto \sumf \left| \sum{\alpha} \langle \nuf | S{\rm int} | \nu{\alpha} \rangle \langle \nu{\alpha} | S_{\rm prop} | \nu_e \rangle \right|^2. \tag{17}$$

We can manipulate this expression into a more convenient form as follows,

$$\sum{f} \left| \sum{\alpha} \langle \nu{f} | S{\text{int}} | \nu{\alpha} \rangle \langle \nu{\alpha} | S{\text{prop}} | \nu{e} \rangle \right|^{2}$$

$$= \sum{\alpha\beta} \sum{\gamma\lambda} (S{\text{prop}}){\alpha\gamma} \rho{\gamma\lambda}^{e}(0) (S{\text{prop}}){\beta\lambda}^{*} \sum{f} (S{\text{int}}){f\beta}^{*}(S{\text{int}}){f\alpha}$$

$$\equiv \sum{\alpha\beta} \rho{\alpha\beta}^{e}(L) \left( \frac{d\zeta}{dE{r}} \right){\beta\alpha} = \text{Tr} \left[ \rho^{e}(L) \frac{d\zeta}{dE_{r}} \right], \qquad (18)$$

where (Si)αβ = ⟨να|Sⁱ |νβ⟩ and we have used the fact that P α |να⟩⟨να| = 1. The term ρ e γλ(0) = ⟨νγ|νe⟩⟨νe|νλ⟩ is the density matrix element describing the initial neutrino system at production. The quantity dζ/dE^r is the generalised scattering cross section, which accounts for the correlation between different neutrino flavour scattering processes. The elements of this quantity can be written as,

$$\left(\frac{\mathrm{d}\zeta}{\mathrm{d}Er}\right){\alpha\beta} = \sum{f=e,\mu\,\tau} (S{\mathrm{int}}){f\alpha}^* (S{\mathrm{int}}){f\beta} \propto \mathcal{M}^*(\nu\alpha \to \nuf) \mathcal{M}(\nu\beta \to \nu_f),$$ (19)

where the two matrix elements, ^M∗ (ν^α → ν^f ) and M(ν^β → ν^f ) describe the processes ναe ^{− → νf e −} and νβe ^{− → νf e −}, respectively.

Finally, the neutrino density matrix ρ e (L) at the detector after propagation can be written as:

$$\rho^{e}(L) = S{\text{prop}} \rho^{e}(0) S{\text{prop}}^{\dagger}$$

$$= e^{-iHL} \begin{pmatrix} 1 & 0 & 0 \ 0 & 0 & 0 \ 0 & 0 & 0 \end{pmatrix} e^{iHL} ,$$ (20)

or, if labelling the matrix elements,

$$\rho{\alpha\beta}^{e}(L) = (S{\text{prop}}){\alpha e}(S{\text{prop}})_{\beta e}^{*}. \tag{21}$$

Here, H represents the Hamiltonian of propagation (see below) while L is the distance travelled by the neutrinos. The resulting density matrix after oscillation is no longer diagonal. Physically, this implies that the incoming neutrino at the detector before scattering is in a superposition of flavour states.

1. Oscillation probability with NSIs

The inclusion of NSIs modifies the vector potential influencing neutrino oscillations in matter, cf. Eq.(9). The Hamiltonian for neutrino propagation from the core of the Sun to the Earth in the flavour basis becomes,

$$H = \frac{1}{2E{\nu}} U \begin{pmatrix} 0 & 0 & 0\ 0 & \Delta m{21}^{2} & 0\ 0 & 0 & \Delta m{31}^{2} \end{pmatrix} U^{\dagger} + \sqrt{2}G{F}N{e}(x) \begin{pmatrix} 1 + \epsilon{ee}(x) & \epsilon{e\mu}(x) & \epsilon{e\tau}(x)\ \epsilon{e\mu}^{*}(x) & \epsilon{\mu\mu}(x) & \epsilon{\mu\tau}(x)\ \epsilon{e\tau}^{}(x) & \epsilon_{\mu\tau}^{}(x) & \epsilon_{\tau\tau}(x) \end{pmatrix} . \quad (22)$$

The ϵαβ are related to the NSI free parameters ε f αβ defined in Eq.(15) as follows3

$$\epsilon{\alpha\beta}(x) = \varepsilon{\alpha\beta}^e + (2\varepsilon{\alpha\beta}^u + \varepsilon{\alpha\beta}^d) + \frac{N_n(x)}{Ne(x)} (\varepsilon{\alpha\beta}^u + 2\varepsilon_{\alpha\beta}^d),$$ (23)

³ Note that we take all ε parameters to be real-valued in this study, although in general they may be complex-valued.

where Nn(x) is the neutron number density, and we have implicitly assumed the proton number density Np(x) = Ne(x) for neutral matter. The above expression is already appropriate for describing NSIs in the heavy mediator limit. However, since the flavour conversion of neutrinos in the presence of matter gets affected by the coherent forward neutrino-matter scattering, we work in the q ^{2 →} 0 limit when accounting for the effect of NSIs on oscillations. This makes it possible to use the same expressions also in the light mediator parameterisation we simply need to translate (gZ′ , mZ′ ) into an equivalent set of (flavour-conserving in our case) ε αα parameters as follows,

$$\varepsilon{\alpha\alpha}^f = \frac{g{Z'}^2 Q{Z'}^f Q{Z'}^{\nu_{\alpha}}}{\sqrt{2} GF m{Z'}^2} \,. \tag{24}$$

Working in the One Mass Dominance (OMD) approximation (∆m^{2 31} → ∞), the density matrix elements can be calculated using an effective 3-flavour oscillation scheme. The (Sprop) in equation Eq.(20) is given by,

$$(S{\text{prop}}){\alpha e} = O{e1}[O{\alpha 1}S{11}^{(2)} + O{\alpha 2}S{21}^{(2)}] + O{\alpha 3}O{e3}e^{-i\Delta{33}L}.$$ (25)

More details about how S (2) ¹¹ and S ²¹ are calculated can be found in Appendix A.

After taking the average over the distribution of production locations in the Sun (which differs for each neutrino flux), the density matrix elements for general flavour indices α, β ∈ {e, µ, τ} become,

$$\rho{\alpha\beta}^{e}(L) = (O{e1})^{2} [(O{\alpha1}O{\beta1})P{ee}^{2\nu} + (O{\alpha2}O{\beta2})P{\text{ext}}

• (O{\alpha1}O{\beta2})P{\text{int}} + (O{\alpha2}O{\beta1})P{\text{int}}^{*}]
• (O{\alpha3}O{\beta3})(O_{e3})^{2}.$$ (26)

The terms O ≡ R23R13, P 2ν ee , Pint and Pext are defined as:

$$O = \begin{pmatrix} c{13} & 0 & s{13} \ -s{13}s{23} & c{23} & c{13}s{23} \ -s{13}c{23} & -s{23} & c{13}c{23} \end{pmatrix}, \tag{27}$$

$$P{ee}^{2\nu} = \frac{1}{2} \left( 1 + \cos(2\theta{12}) \langle \cos(2\theta_m) \rangle \right) ,$$ (28)

$$P_{\rm int} = -\frac{1}{2} \langle \cos(2\thetam) \rangle \sin(2\theta{12}) e^{i\delta_{\rm CP}} , \qquad (29)$$

$$P{\text{ext}} = \frac{1}{2} \left( 1 - \cos(2\theta{12}) \langle \cos(2\theta_m) \rangle \right) , \qquad (30)$$

where sij = sin θij , cij = cos θij , θij are the vacuum mixing angles and δCP is the CP-violating phase. We use the best fit values given in ref [141]. θ^m is the effective mixing angle in matter, calculated by finding the eigenvalues of the Hamiltonian in Eq.(22) in the OMD approximation, detailed further in Appendix A. The terms containing

factors of e ⁱ∆33^L rapidly oscillate as a function of neutrino energy, and the resulting density matrix elements after averaging do not depend on them.

The electron flavour diagonal density matrix component is calculated to be,

$$\rho{ee}^{e}(L) = s{13}^{4} + c{13}^{4} P{ee}^{2\nu} = P_{ee}^{3\nu}. \tag{31}$$

Hence, the diagonal terms of the density matrices, ρee, ρµµ and ρτ τ correspond to the measurement probabilities of the system, i.e. Pee, Peµ and Peτ respectively. These will equate to the SM case, when all ϵαβ parameters in Eq.(22) are set to zero.

2. Cross section

For EνeS we set f = e in the Lagrangians of Eqs.(13) and (14). In this work, we consider purely vector couplings for which g^L = gR. The effects of the light mediator NSI in the cross section can be included by modifying the couplings in Eq.(11) [62, 71]:

Light mediator: (32)

$$g{L/R}^{\rm NSI} = g{L/R}^{\rm SM} + \frac{g{Z'}^2 Q{Z'}^e Q{Z'}^{\nu{\alpha}}}{2\sqrt{2} G_F \left(2 m_e Er + m{Z'}^2\right)} \,. \label{eq:nsi}$$

where the charges Q are chosen from Table I. In this case, it is readily apparent that we recover the SM when we switch off NSIs by setting gZ′ = 0.

For the heavy mediator case, to account for flavourviolating effects, we replace the SM cross section with the generalised scattering cross section [46, 71, 140],

Heavy mediator: (33)

$$\begin{split} &\left(\frac{\mathrm{d}\zeta}{\mathrm{d}Er}\right){\alpha\beta}^{\mathrm{NSI}} = \frac{2G_F^2 m_e}{\pi} \sumf \left[G{\alpha f}^L G{f\beta}^L \right. \ &\left. + G{\alpha f}^R G_{f\beta}^R \left(1 - \frac{E_r}{Ev}\right)^2 - \left(G{\alpha f}^L G{f\beta}^R + G{\alpha f}^R G_{f\beta}^L\right) \frac{m_e Er}{2E\nu^2}\right], \end{split}$$

where

$$G{\alpha\beta}^{L} = g{L\beta}\delta{\alpha\beta} + \varepsilon{\alpha\beta}^{e,L}, \quad G{\alpha\beta}^{R} = g{R}\delta{\alpha\beta} + \varepsilon{\alpha\beta}^{e,R}.$$ (34)

If we turn off NSIs by setting ε e,L αβ and ε e,R αβ to zero, the generalised cross section becomes diagonal, and the diagonal components dζ dE^r ee dζ dE^r µµ and dζ dE^r τ τ correspond to the SM cross sections ^dσ^e dE^r dσ^µ dE^r and ^dσ^τ dE^r of Eq.(11) respectively. Note that the label αβ does not mean that α is the initial and β the final neutrino state they are both initial-state indices, as shown in Eq.(19).

3. Double-differential event rate with NSI

A change of neutrino flavour during a scattering process is disallowed in the SM. In terms of Eq.(19), this

FIG. 1. Yearly expected number of electron recoil events with energies E^r > 10 keV, in a 30 m³ gas detector containing a 60:40 mixture of He and CF⁴ at atmospheric pressure. Each line shows the total rate as a function of a different ε e αβ where all other values of ε e αβ are kept at zero. The case for the Standard Model is shown with a dotted grey line. The thick part of each line shows the range of ε allowed by Borexino at 90% CL. The event numbers shown here correspond to the sum over the pp and ⁷Be fluxes.

means that the final neutrino state f should be the same as the initial neutrino states α and β; i.e f = α = β. This implies that the generalised cross section is diagonal, and as seen in Sec. II C 2, these diagonal elements correspond to the standard model cross sections. We also saw in Sec. II C 1 that the diagonal elements of the density matrix ρ e correspond to the transition probabilities Peα. Thus, if we were to express our SM event rate in terms of the density matrix and generalised cross section, it would reduce to a product of transition probability and cross section:

$$\begin{split} \left(\frac{\mathrm{d}^2 R}{\mathrm{d} E_r \mathrm{d} \Omega_r}\right)^{\mathrm{SM}} &\propto \mathrm{Tr} \left[ \rho^e \left(\frac{\mathrm{d} \zeta}{\mathrm{d} Er}\right) \right]^{\mathrm{SM}} \ &\propto \sum{\alpha \beta} \left[ \rho^e_{\alpha \beta} \left(\frac{\mathrm{d} \zeta}{\mathrm{d} Er}\right){\beta \alpha} \delta{\alpha \beta} \right] = \sum{\alpha} P{e\alpha} \frac{\mathrm{d} \sigma{\alpha}}{\mathrm{d} E_r} \end{split}$$

However, with NSI effects included, we can have f ̸= α ̸= β which means that the generalised cross section can have off-diagonal terms. To account for all these processes, we must express the NSI-modified event rate as,

$$\left(\frac{\mathrm{d}^2 R}{\mathrm{d} E_r \mathrm{d} \Omega_r}\right)^{\mathrm{NSI}} \propto \mathrm{Tr} \left[ \boldsymbol{\rho}^e \left(\frac{\mathrm{d} \boldsymbol{\zeta}}{\mathrm{d} E_r}\right) \right]^{\mathrm{NSI}}.$$ (35)

This expression is therefore a more general expression for event rate calculations—it can be used for both SM flavour-conserving interactions, as well as NSI flavourviolating interactions.

So in summary, we rewrite the full expression for the double-differential event rate that we wrote down in Eq.(8) in terms of a density matrix and generalised cross section,

$$\frac{\mathrm{d}^2 R}{\mathrm{d} E_r \mathrm{d} \Omega_r} = \frac{N_T}{2\pi} \frac{\mathcal{E}^2}{y} \left( \mathrm{Tr} \left[ \rho^e \frac{\mathrm{d} \boldsymbol{\zeta}}{\mathrm{d} Er} \right] \frac{\mathrm{d} \boldsymbol{\Phi}}{\mathrm{d} E{\nu}} \right) \bigg|{E{\nu} = \mathcal{E}} . \quad (36)$$

Finally, we note that we must also calculate this rate for each solar neutrino flux separately, as opposed to defining Φ = Φpp + Φ⁷Be + ... as the sum of all fluxes. This is because the density matrix depends on the neutrino production profiles within the Sun, which are different for each flux. For electron recoils, we consider both pp and ⁷Be fluxes, with pp contributing around 75% of the rate. The remaining solar fluxes contribute less than a handful of events, given the detector volumes we are interested in here. For nuclear recoils (Appendix B), we consider only the ⁸B flux. For the full electron and nuclear recoil rates across all of the solar neutrino fluxes, as well as prospects for their detection and spectroscopy using directional gas detectors, we refer to our previous study Ref. [127].

D. Comparison of SM & NSI total rates and angular recoil spectra

In general, the inclusion of nonzero ε e αβ leads to an enhancement in the cross section and so a larger event rate compared to the SM prediction, roughly scaling as R ∝ (ε e αβ) . That said, it is possible for interference between positive and negative terms in the cross section, leading to a suppression in the rate relative to the SM.

In Fig. 1, we display the total expected number of electron recoils per year in a 30 m³ gas volume containing a mixture of He and CF⁴ at atmospheric pressure. This choice of volume and gas mixture is motivated further in the next section—for now, we focus on the dependence of the total rate on the NSI parameters, where Fig. 1 applies to the heavy mediator limit. Here we consider varying one NSI parameter at a time, with all others fixed at zero, including ε u,d αβ . We see that, in general, turning the NSI parameters on leads to an enhancement in the total rate. The most obvious exception is ε e ee ≈ −1.2 for which there is a partial cancellation in the cross section. Cancellations also occur for the other couplings but are very small in magnitude and occur for small values of ε, so they are not easily visible on the plot. For comparison, we have highlighted the current range of ε e αβ allowed at 90% CL, from an analysis of Borexino's Phase-II spectral data [71], under the same assumption that only one NSI coupling is active at a time.

From Fig. 1, we can already conclude that while the event numbers are at an observable level in an experiment of this scale, obtaining statistics needed to discriminate against the SM for ε e αβ ^{≲ 0}.5 will require either a substantial increase in exposure (volume × time) or an extremely low electron recoil background. However, as we will motivate now, the ability of MPGD-based experiments to reconstruct both the recoil energy and the recoil direction independently will greatly alleviate the required background level.

To highlight the effect of the NSI parameters on the energy-angle distribution, Figs. 2 and 3 display the double-differential event rates. In this case, we are showing the double-differential event rate in heliocentric coordinates, i.e. in terms of cos θ⊙, where cos θ^⊙ = 1 corresponds to a recoil pointed in the direction of the solar vector, ˆq⊙. The side panels of Fig. 2 also show the differential event rate as a function of cos θ^⊙ and E^r alone, i.e. after integrating d2R/dErd cos θ^⊙ over energy and angle respectively. Both panels in Fig. 2 demonstrate that the recoil distribution remains pointed towards the solar direction. Recall that energy and angle are correlated: the highest energy recoils carry more of the incoming neutrino's forward momentum, so they will have small scattering angles or values of cos θ^⊙ close to 1. When NSIs are turned on, the event rate is generally enhanced more at low energies than at high energies, which means the angular distribution becomes slightly less focused towards the solar direction. We show two illustrative cases, one for the heavy mediator limit and one for the B − L light mediator, but we have checked many different configurations and find that the behaviour seen in Fig. 2 is generic.

To illustrate this latter point further, Fig. 3 shows the relative difference between the SM and the heavy mediator NSI cases, again when each coupling ε e αβ is turned on, while all others are fixed at zero. The blue colours show regions of this space of observables where the NSIs lead to a suppression in the rate compared to the SM, while pink regions are where it receives an enhancement. We note that while the general behaviour is the same, there are subtle differences in the effects of each set of αβ on d ²R/dErd cos θ⊙. These differences are not present at the level of the total event rate (Fig. 1), where turning on different ε e αβ couplings can lead to identical values for the total expected number of events. This result motivates the use of a detector able to measure both E^r and cos θ^⊙ to attempt to search for the presence of NSIs and identify which interactions may or may not be active.

III. DIRECTIONAL ELECTRON-RECOIL DETECTION WITH GAS TPCS

We now describe how we model a directional gas-based detector. The first quantity to fix is the target medium. In line with the promising simulations and sensitivity studies performed in the context of the CYGNO experiment, we will assume the use of atmospheric pressures of He:CF4. This gas mixture is an electron drift gas and can allow high gas gains of ^O(10⁶ ) with gas electron multipliers (GEMs), at the minor cost of worse diffusion compared with negative-ion drift gases like SF6. At the relevant energies, electron recoils have much longer tracks than nuclear recoils, but lower ionisation densities, which means enhancing signal-to-noise is preferable over spatial resolution. The CF⁴ gas also scintillates in the visible range, which as well as enabling the use of an optical readout (i.e. cameras), also means the light can be timed to use for reconstruction of a track's angle with respect to the drift direction orthogonal to the readout. This is the strategy for full 3-dimensional spatial reconstruction adopted by CYGNO.

The addition of helium to the CF⁴ has many advantages. In particular, it provides more scattering sites without substantially increasing the gas density, and so degrading the directional performance. It also allows the detector to operate at atmospheric pressure, simplifying the design and engineering of the gas vessel.

In Ref. [127], we motivated the use of a 60:40 ratio for He:CF⁴ at atmospheric pressure for solar neutrino detection. Since preparations are underway by the CYGNO collaboration for a 30 m³ scale detector using this gas mixture, we will fix 30 m^{3 ×} 5 years as a near-term exposure benchmark, but will entertain further scale-ups if competitive sensitivity to some physics channel requires it. Since ongoing design studies are now focused primarily on materials screening and background reduction, our discussion will primarily focus on the impact of the total background rate Rbg as opposed to total volume. Discussion of our assumed background model can be found below in Sec. III B.

A. Detector performance

Having derived the input theoretical double-differential event rate d²R/dErdΩ^r in Sec. II, we now describe how these event rates can be converted into functions that more closely resemble the space of observables for a gasbased time-projection chamber.

The two main quantities a direction-sensitive detector aims to reconstruct are the total recoil energy and the initial recoil direction. This is done by reconstructing the three-dimensional ionisation cloud produced when the recoil deposits energy as it slows down in the detector medium. The full three-dimensional ionisation distribution is typically obtained by combining a 2-dimensional projection onto the highly pixelated MPGD readout, with the third axis relative to that plane reconstructed using timing. Depending on the type of readout involved, the arriving charge can either be detected directly (electronic readout) or the scintillation produced during the amplification stage can be imaged with high-resolution cameras (optical readout). A detailed understanding of the expected ionisation distributions from recoils of a given energy and particle type is then required to infer the initial energy and direction from the reconstructed

FIG. 2. Two illustrative cases of the double-differential event rates as a function of recoil energy, Er, and recoil angle with respect to the Sun, cos θ⊙. The central coloured panels display d²R/dErd cos θ^⊙ for the NSI case. We assign a set of ε e αβ as labelled for the heavy mediator example (left), and (mZ′ , gZ′ ) as labelled for the B − L light mediator example (right). The smaller panels to the left and above the main panel in each case show the event rate integrated over angle and energy, respectively. In all panels, the solid lines correspond to the NSI case while the dashed lines show the SM (∀ε e αβ = 0 or gZ′ = 0 for heavy and light mediators, respectively). In all of these cases, the lines correspond to the rate after it has been normalised by the maximum value so as to highlight differences in shape (although the total rate also differs, as shown in Fig. 1). In the main panels, the solid and dashed contours also delineate equal values of the rate normalised by the maximum value—the four levels correspond to 0.75, 0.5, 0.25 and 0.1.

ionisation track. This procedure involves either fitting the distributions using traditional methods, or using dedicated machine-learning algorithms [87, 112]. Since we are not performing full gas simulations here, our goal is to write down simple functional descriptions of the physical effects that limit the ability to reconstruct the initial recoil direction and energy perfectly, inspired by the performance of existing algorithms tested on experimental data. We do this following the same strategy as in Ref. [127], which we recap briefly here.

We emulate the performance of the detector by convolving the underlying theoretical event rate with several performance kernels. These describe the major effects that degrade the ability of the detector to reconstruct the correct energy and cos θ. These effects are especially important at low energies when the tracks become small and signal-to-noise decreases. The most efficient approach to apply these effects is at the event level, by running a Monte Carlo simulation. The procedure we follow is described further in Ref. [127], in short, it amounts to first generating recoil energies and angles from d²R/dErd cos θ⊙, and then resampling those energies and angles from a set of kernels that describe the energy and angular resolutions. The final step is then to randomly assign a vector sense to the recoil direction according to a head-tail efficiency function. We describe these functions below. Then, when those effects are applied, we re-bin the events into a new 'observed' doubledifferential event rate, d²Robs/dErd cos θ⊙.

The effect of finite energy resolution is captured by a Gaussian with width σ^E that depends on energy. Given a true underlying value of E^r we draw a 'measured' recoil energy E′ r from the distribution,

$$K{E}(E{r}, E{r}') = \frac{1}{\sqrt{2\pi}\sigma{E}(E{r})} \exp\left(-\frac{(E{r} - E{r}')^{2}}{2\sigma{E}^{2}(E_{r})}\right).$$ (37)

We use the following simple function for the energy spread,

$$\sigma_E(E_r) = pE \sqrt{\frac{E{\text{ref}}}{E_r}} \,. \tag{38}$$

which has been well-fit to both simulated and real electron recoils in gas at tens to hundreds of keV by the CYGNO collaboration [142]. We adopt p^E = 0.03 and Eref = 35 keV for our baseline performance, which is inspired by taking an optimistic view of these experimental results, as discussed in Ref. [127]. The reference energy of 35 keV is arbitrary, but here we choose it to lie at the 10th percentile of the recoil energy distribution above the threshold of 10 keV, which means we can interpret the value of p^E as the resolution achieved by 90% of the recoil spectrum.

FIG. 3. Similar to Fig. 2, only now we show the relative difference in the energy-angle distributions of the recoils, which quantifies the differences compared to the SM for all six of the NSI parameters. More precisely, we are showing (rNSI − rSM)/rSM where r = (1/R)d²R/dErd cos θ^⊙ is the double differential event rate normalised by the total rate. For each panel, we set all other ε e αβ to zero apart from the component labelled, which we set equal to 1. This figure highlights that ε e ee, ε e eµ, and ε e eτ have the largest relative impact on the angular distributions.

Next, the reconstruction of cos θ is degraded by an angular resolution kernel taken to be a von Mises-Fisher distribution [112],

$$K_{\theta}(\hat{\mathbf{q}}, \hat{\mathbf{q}}', E_r) = \frac{\kappa(E_r)}{4\pi \sinh \kappa(E_r)} \exp(\kappa(E_r)\mathbf{q} \cdot \mathbf{q}') , \quad (39)$$

where the parameter that specifies the variance of the von Mises-Fisher distribution is κ. Rather than quoting this parameter, we will express the angular resolution in terms of the more intuitive variable, the root-meansquared axial angular resolution σθ, which is a monotonic function of κ that can be evaluated numerically. We find this relationship by calculating the root-mean-square axial angle between a sample of recoil vectors generated according to the distribution above, and then repeating this process over a suitable range of κ. The von Mises-Fisher distribution has the advantage over, for example, a truncated Gaussian, in that it reflects the fact that the distribution exists on the sphere.

The root-mean-squared angle between two axes approaches σ^θ = 1 radian for completely uncorrelated directions (i.e. the limit of no directional sensitivity). Following on from the discussion of σE(Er) above, we approximate the energy-dependence of the angular resolution

with the following function form,

$$\sigma_{\theta}(Er) = p{\theta} \sqrt{\frac{E_{\text{ref}}}{E_r}},$$ (40)

where p^θ = 10◦ for our baseline detector performance benchmark, as inspired by results from gas simulations of electron recoils at the relevant energies [142, 143]. Note that the E^r appearing in this equation is the true recoil energy, not the reconstructed energy, E′ r .

One important assumption that we should emphasise here is that, by taking cos θ^⊙ as our derived measurable quantity, we are assuming that the detector has threedimensional reconstruction, and so the track direction can be rotated into heliocentric coordinates. Although we do this for simplicity (relaxing this assumption would force us to also consider the event time and detector orientation), the ability to perform three-dimensional direction reconstruction is considered to be a major design feature in all MPGD-based detectors being developed right now. CYGNO is able to achieve this by first measuring the two-dimensional projection of the recoil track along the optical readout plane made up of highly sensitive CMOS cameras, which are placed beyond an amplification stage made up of GEMs. This projection is then combined with the one-dimensional projection along the drift axis provided through the timing of scintillation light detected by photomultiplier tubes. The combination of these signals is what also enables the precise reconstruction of the recoil energy.

The final detector performance effect we account for is the limitation in reconstructing the correct ± sense of the recoil direction. This is a particularly important effect at the lowest energies, where tracks are shortest and the ionisation density is low. It is possible at these energies that the track alignments could be reconstructed even while the head and the tail of the track are not. The head and tail of a recoil are primarily reconstructed by using the differences in the dE/dx along the track. Unlike nuclear recoils, which are on the stopping side of the Bragg peak, the electrons deposit more ionisation as they slow down. So the tail (end) of the track curls up and contains the most ionisation. Since in our simplified description, headtail assignment is binary, we simply randomly assign a ± sign to each recoil vector with a probability given by the function,

$$\varepsilon_{\text{HT}}(Er) = \frac{1}{2} \left( \frac{1}{1 + e^{-a{\text{HT}}(Er - E{\text{HT}}))}} + 1 \right).$$ (41)

This functional form captures the behaviour of head-tail performance well for all recoils, which, by definition, must begin at zero at E^r = 0, and plateau at 1 for high-enough energies. For our baseline performance we assume aHT = 0.2 and EHT = 20 keV. Since the neutrino recoils are restricted to a single hemisphere due to the scattering kinematics (up to the effects of angular resolution), the primary effect of the head-tail performance is to limit the ability to distinguish neutrino events from isotropic backgrounds.

The final effect we should include in principle is the event-level efficiency of the experiment. This is more difficult to quantify at present, as it will depend on the specifics of a given detector and the way the data is collected and analysed. To keep the discussion in this study as clear as possible for when our results are applied in real detector simulations by the CYGNO collaboration, we simply incorporate a step-function efficiency curve by imposing a hard threshold at 10 keV. Our results are relatively weakly dependent on this threshold since the electron recoil distribution due to pp and ⁷Be neutrinos is not very steeply falling at these energies, and in any case, all of our energy and angular performance curves have already degraded significantly by this energy.

B. Backgrounds

The rate of non-neutrino electron-recoil backgrounds in experiments of this type is the main quantity that will limit their sensitivity. Fortunately, the directionality will provide an excellent technique to reject backgrounds, which are expected to be close to isotropic. Nevertheless, backgrounds will appear in the signal region defined by cos θ^⊙ ∈ [0, 1], so we must incorporate some background model that quantifies this.

We make an educated guess as to the rate and energy spectrum of the background using recent background simulations performed in preparation for a future 30 m³ scale-up of the CYGNO experiment to be located at LNGS. Note that the general design parameters of CYGNO-30, in terms of gas mixture and directional performance, have already inspired most of the choices we have made in this study so far. The differential recoil spectrum for an isotropic background model can be written as follows,

$$\frac{\mathrm{d}^{2} R{\mathrm{bg}}}{\mathrm{d} E{r} \mathrm{d} \Omega{r}} = \frac{R{\mathrm{bg}}}{4 \pi E{\mathrm{bg}} \left( e^{-E{\mathrm{th}}/E{\mathrm{bg}}} - e^{-E{\mathrm{max}}/E{\mathrm{bg}}} \right)} e^{-E{r}/E_{\mathrm{bg}}}. \tag{42}$$

For our chosen model we have Eth = 10 keV and Emax = 700 keV. The model is parameterised by one free parameter, the total background rate Rbg. We fix the scale of the exponential fall off of the background energy spectrum to be Ebg = 67 keV, which we find to be a good fit to the spectrum of the total internal background from a simulation of CYGNO-30 [142]. This simulation includes all material electron backgrounds in the 0–1000 keV energy window, with the field cage being the dominant contributor to the background budget, followed by the GEMs and camera sensors.

As for the amplitude of the background rate, Rbg, to make the discussion more intuitive, we will express its value relative to the total event rate from solar neutrinos R^ν under the SM (see the grey dashed line in Fig. 1). Importantly, the simulations for CYGNO [142] suggest Rbg/R^ν ∼ 500 is already achievable without any dedicated background reduction which, as we will discuss,

is quite promising. In particular, this study did not account for the possibility of using the ultra-low radioactivity kapton presented in Ref. [144], and so naively a further background reduction is possible already. To see the impact of the size of the background on the scientific reach of the experiment we will explore a wide range of possible background reduction factors, while keeping in mind that Rbg = 50R^ν is a realistic expectation for a fully-deployed experiment of 30 m³ volume. To demonstrate the power of directionality, in our likelihood analysis described in the next section, we will float the value of Rbg as an unknown nuisance parameter.

IV. LIKELIHOOD ANALYSIS

Having described all the necessary ingredients for calculating a reasonable approximation to the neutrino signal detectable by a gas-based directional detector, the final step is to develop a statistical methodology. Our previous analysis in Ref. [127] focused on the ability of the detector to measure the neutrino flux over some experimental background. Here, our scientific question is slightly different. Given an observed set of neutrinoinduced electron recoil energies and directions, is their distribution consistent with the SM or not? Fortunately, the SM event rate can be expressed as a special case of the more general NSI formalism (i.e. ∀ε e αβ = 0, or gZ′ = 0) which means our null hypothesis that the SM is true is nested within a more general alternative hypothesis that the event rate is described by Eq.(35). This then permits us to use the profile likelihood ratio test and apply Wilks' theorem.

We start by writing down a likelihood. To facilitate the use of Asimov statistics (see below), we will use a binned likelihood based on the event rate having been binned in energy and cos θ⊙, i.e.,

$$N_{\nu}^{ij}(\varepsilon) = MT \times \tag{43}$$

$$\int{\cos\theta{\odot}^{i}}^{\cos\theta{\odot}^{i} + \Delta\cos\theta} \int{E{r}^{f}}^{E{r}^{j} + \Delta E{r}} \frac{\mathrm{d}^{2}R{\mathrm{obs}}(\varepsilon)}{\mathrm{d}E{r} \mathrm{d}\cos\theta{\odot}} \mathrm{d}E{r} \, \mathrm{d}\cos\theta{\odot}$$

where MT is the exposure (mass × duration) of the experiment. Our bins run over the domain cos θ^⊙ ∈ [−1, 1] and E^r ∈ [10, 700] keV across 50 linearly spaced bins each. The exact binning scheme does not impact our results since the finite energy and angle reconstruction are already taken care of, as discussed in Sec. III. For compactness, we use the shorthand here ε ≡ {ε e αβ} to denote the set of all ε values upon which we are performing the statistical test. The background distribution is constructed using the same binning scheme. We parameterise the background using a normalisation b whose correct value is one. This means we can parameterise an expected number of events in bin ij as,

$$N{\rm exp}^{ij}(\varepsilon, b) = N{\nu}^{ij}(\varepsilon) + bN_{\rm bg}^{ij}$$ (44)

FIG. 4. Projected median 90% CL exclusion limits on each ε αβ parameter (assuming all other ε e αβ = 0). We assume a detector of 30 m³ volume with a 60:40 He:CF⁴ gas mixture at atmospheric pressure. The three panels from left to right are for different background rates, measured relative to the total rate of neutrino-induced electron recoils under the SM, Rν. The various projected limits shown in blue are for different levels of directional sensitivity: idealised performance, realistic performance, and no directionality, from lightest to darkest. We emphasise that the "non-directional" case has an identical event rate, energy threshold and energy resolution as the "directional" cases. For comparison, in black we show the existing constraints from Borexino, demonstrating a competitive level of sensitivity despite the small size of the proposed experiment.

We now construct a binned likelihood for the parameters {ε, b} given some observed data N ij obs which we assume to be Poisson distributed with mean Nij exp,

$$\mathcal{L}(\varepsilon, b) = \prod{i} \prod{j} \text{Poiss} \left[ N{\text{obs}}^{ij}; N{\text{exp}}^{ij}(\varepsilon, b) \right]. \tag{45}$$

In practice, we use the log-likelihood, which can be written as,

$$\ln \mathcal{L}(\varepsilon, b) = \sum{ij} \left( N{\text{obs}}^{ij} \ln[N{\nu}^{ij}(\varepsilon) + bN{\text{bg}}^{ij}] - (N{\nu}^{ij}(\varepsilon) + bN{\text{bg}}^{ij}) \right)$$ (46)

we neglect the Nobs! term which will cancel when we take likelihood ratios.

We wish to construct a profile-likelihood ratio test that quantifies the extent to which the null hypothesis of the SM, i.e. ε = 0, describes the data. To do this, we will assume that the observed data is described by the SM and forecast the expected exclusion limits one could then set on ε. We write down the standard profile likelihood ratio test statistic used for deriving exclusion limits as [145],

$$TS(\varepsilon) = -2 \left( \ln \mathcal{L}(\varepsilon, \hat{b}) - \ln \mathcal{L}(0, \hat{\hat{b}}) \right)$$

where ˆb is the maximum likelihood estimator (MLE) at some fixed value of ^ε, whereas ˆˆ^b is the MLE when ^ε is fixed to its value under the null hypothesis. Wilks' theorem holds in this model setup, which states that the

distribution of TS asymptotically follows a χ ⁿ distribution where n is the number of parameters in ε.

To make an estimate of the expected exclusion limit that a future experiment could set, we employ the Asimov dataset. The Asimov dataset is a hypothetical situation in which the observed data exactly follows the expectation under some assumed set of "true" parameter values. Since we are projecting exclusion limits, where we imagine that the SM hypothesis correctly describes the data, we set Nobs = Nexp(∀ε = 0, b = 1). The MLE for the background parameter under the null hypothesis is then trivially ˆˆ^b = 1, whereas the MLE for the alternative hypothesis can be found numerically by solving,

$$\frac{\partial \ln \mathcal{L}(\varepsilon, b)}{\partial b} = \sum{ij} \left( \frac{N{\rm obs}^{ij} N{\rm bg}^{ij}}{N{\nu}^{ij}(\varepsilon) + b N{\rm bg}^{ij}} - N{\rm bg}^{ij} \right) = 0. \quad (47)$$

As described in Ref. [145], the value of TS under the Asimov data set asymptotically approaches the median of its distribution. That is to say, when we forecast an exclusion limit using the Asimov data set, we find a good estimate of the level of sensitivity that at least 50% of hypothetical experiments could achieve. To forecast a limit at a given CL we then simply need to find values of ε at which the TS crosses some threshold value. For setting two-sided limits at the 1−p level on the n parameters in ε we use the 1−p percentile function of the χ 2 ⁿ distribution. For one-sided limits, e.g. on the parameter g^V in the lightmediator case, we use the 1 − 2p percentile function.

FIG. 5. Projected median 90% CL exclusion limits when two ε e αβ parameters are considered simultaneously (assuming all other ε e αβ = 0). We assume a detector of 30 m³ volume with a 60:40 He:CF⁴ gas mixture at atmospheric pressure. The three columns correspond to three different pairs of NSI parameters. The top set of panels is for a background rate that is fifty times the neutrino electron recoil rate under the SM, while the bottom set of panels is for five times the neutrino rate. The various projected limits shown in blue are for different levels of directional sensitivity: idealised performance, realistic performance, and no directionality, from lightest to darkest. We emphasise that the "non-directional" case has an identical event rate, energy threshold and energy resolution as the "directional" cases.

V. RESULTS

We will now show estimated median 90% exclusion limits on the various ε e αβ and (gZ′ , mZ′ ) parameters for a range of choices for the directional performance and background conditions. Since we are motivated by the upcoming 30 m³ detector being developed by the CYGNO collaboration, we will present results for this fixed total volume, and provide an analytic scaling to understand how to extrapolate our results to larger volumes. Our primary goal is to motivate small-scale direction-sensitive gas detectors in general, as well as to understand the level of electron recoil background that is required to set relevant constraints on the neutrino cross section using a detector of this type.

To have something to compare our results against, it is useful to define what we call an equivalent "nondirectional experiment". This is a hypothetical experiment that observes an identical event rate as our directional experiment, but does not have access to any cos θ^⊙ information. We can forecast limits for this experiment in the same way, but instead of Nij exp appearing in the likelihood, we first take the sum over cos θ^⊙ bins in the equation above, i.e.

$$N{\mathrm{exp,non-dir}}^{j}(\varepsilon) = \sum{i} N_{\mathrm{exp}}^{ij}(\varepsilon) \,.$$ (48)

As another point of comparison, we also define an "idealised performance" benchmark where the energy and angular performances are not included. This amounts to calculating Nexp directly from the underlying d²R/dERd cos θ^⊙ as opposed to d²Robs/dERd cos θ^⊙ which includes the performance effects. All of the figures in this section will then show three benchmark experiments: non-directional, directional (realistic) and directional (idealised)—with the expected sensitivity improving in that order. We reiterate at this point that our goal is not to claim world-leading sensitivity to NSIs, but to highlight that—thanks to directionality—a gas-based experiment can make an interesting contribution to the global picture of the neutrino's properties, despite being several orders of magnitude smaller and cheaper than its

FIG. 6. Projected constraints on a light vector mediator under the U(1)B−^L model, as a function of the coupling gZ′ and mediator mass mZ′ . The solid lines show median 90% CL exclusion limits for the three experimental benchmarks used in the previous figures. The constraint from Borexino from Ref. [71] is shown for comparison. The shaded regions show example 1σ discovery contours if the true values of the parameters were situated at the gold star. We emphasise that the directional information (and the background rejection that comes with it) is what would allow such a discovery claim to be made. Following conventions, the shaded region encloses the excluded parameter space, as opposed to the previous figures, where the shaded regions were the allowed parameter space.

nearest competitors, e.g., Borexino.

First, in Fig. 4 we show a set of forecasted two-sided median 90% CL exclusion limits (TS = 2.71) on various ε e αβ parameters in the heavy mediator limit. These are one-dimensional constraints where we turn on one NSI coupling from αβ = {ee, eµ, eτ, µµ, µτ, τ τ}, with all others set to zero. As a direct point of comparison, we show the two-sided 90% CL exclusion limits from an analysis of Borexino Phase-II spectral data [71]. In all cases, the gas mixture, exposure time, volume, and pressure are fixed as labelled in the figure. The three panels from left to right vary only the background rate, Rbg, from 5 to 500 times the SM neutrino rate. The latter value is the current estimate of the background level from simulations, prior to any dedicated screening or reduction efforts. We see that the constraints for the worst background condition are not at a competitive level, but after only a ×10 reduction, the constraint for certain couplings like ε e eτ reach within a modest factor of Borexino's. For an even more aggressive reduction of a factor of 100, the exclusion limits almost match those of Borexino. The case for directionality in particular is made the most apparent for

ε e ee where the added information provided through the cos θ^⊙ distribution as well as the background rejection power allows for the region around ε e ee ≈ −2—which is mildly degenerate with the SM at the level of the total event—rate to be excluded. Across all cases, we observe that the exclusion limit on ε scales as,

$$\varepsilon \propto (MT)^{-1/4} R_{\rm bg}^{1/4} \,. \tag{49}$$

It is also possible for multiple NSI couplings to be active at once and to interplay with each other at the level of the event rate. To capture this possibility, we show three more examples in Fig. 5. Now we are showing twodimensional 90% CL exclusion limits on two degrees of freedom (TS = 4.61), where two ε e αβ are turned on, with all others fixed at zero. The top and bottom sets of panels are again for two background assumptions: 50× and 5× the neutrino rate, respectively. Again, we see from comparing the darkest blue contours (non-directional) with the two lighter contours that the power of directionality is apparent. It is clearest again in the ee case, where the degeneracy with the SM for large negative values of ε e ee is broken.

Then finally, we come to the light mediator example in Fig. 6. As discussed in Sec. II we are showing only the B − L model here with ∀Q e,ν^{α Z}′ = −1, but have checked that we obtain almost identical results for other models. We adopt a slightly different definition for the exclusion limits here and following the typical convention—here we have forecasted the median one-sided 90% CL exclusion limits on one parameter, gZ′ , with the two-dimensional constraint then built up by repeating the test over a range of values for mZ′ . The Borexino constraint is again from Ref. [71].4 We note that there are various other constraints from stellar cooling, fixed target experiments, and BBN/Neff that are also present for these light mediator models that partially overlap this region. We choose to remove these for the sake of clutter and because our main message is the comparison between these two solar neutrino experiments. We note that it is always possible to model build around any of these constraints, and not all are present for every instance of an anomaly-free gauged U(1)′ model.

As expected, the sensitivity shown in Fig. 6 is best at low mediator masses where the deviation in the cross section due to NSIs emerges at energies that are below Borexino's threshold. Borexino, of course, still leads at higher mediator masses. We show the case here for Rbg = 50 Rν, but sensitivity scales slowly gZ′ ∝ (MT) ⁻1/⁸R 1/8 bg , so the results for 5 R^ν are similar.

For this figure, in addition to the forecasted exclusion limits, we show another way to demonstrate the power

⁴ We note that the Borexino constraint shown here is actually a two-sided 90% CL exclusion on gZ′ rather than the one-sided limit on |gZ′ | as we have shown. The differences between the two definitions are almost invisible on the plot at the scale we have shown it, so we choose not to debate this choice.

of directionality by forecasting a possible discovery. The ability to make discoveries is another of the major motivations behind directional detection that has been appreciated for many years (see e.g. the discussion in the context of dark matter in Ref. [87]). Although experiments like XENONnT, LZ and PandaX have the statistics to set constraints at a similar level to what is shown here in principle, in practice, it is challenging for them to make positive claims of a signal due to the high electron background preventing the robust identification of neutrino events. This is not the case for a directional experiment, where a much stronger statement about the probability of any given recoil as being due to a neutrino is possible, thanks to the angular domain being split into an (almost) pure background hemisphere and a signal+background hemisphere. To showcase this we show on Fig. 6 some example median 1σ discovery contours, given some arbitrarily chosen hypothetical "true" value for (gZ′ , mZ′ ) = (g true ^Z′ , mtrue ^Z′ ) shown by the gold star. We calculate this contour using a "discovery" test statistic, which essentially amounts (in the Asimov procedure) to fixing Nobs = Nν(g true ^Z′ , mtrue ^Z′ ) + Nbg, whereas previously for exclusion limits we set g true ^Z′ = 0. Comparing the resulting contours between the directional and nondirectional examples, we again see the benefit that comes with directionality.

VI. CONCLUSIONS

In this paper, we have motivated the continued development of 30 m³ -scale gaseous detectors for the directional detection of solar neutrinos through electron recoils. In alignment with the message of Ref. [127], we argue that an experiment of this type can perform neutrino physics, trading in cost and overall size for the added signal information and background rejection that comes from directional sensitivity. We illustrated this in the context of the neutrino NSI formalism, where the interaction between neutrinos and electrons is modified by the presence of a new vector Z ′ mediator beyond the SM.

Our results are illustrated in Figs. 4 and 5 in the heavy mediator limit (mZ′ ≫ q), where we permit both flavourconserving and flavour-violating interactions. Then, in Fig. 6, we showed the case for a light mediator, where we allow for only flavour-preserving interactions, as expected in the gauged U(1)B−^L model. We have not attempted an exhaustive study of all possible combinations of NSI flavour structures, alternative mediator types, or other U(1)′ models, as the results presented here are sufficient to convey our main message. We believe a background reduction of around a factor of ten is needed from preliminary detector background simulations to reach an interesting level of sensitivity to neutrino interactions beyond the SM using solar neutrinos.

Our results remain to be validated in a full Monte Carlo simulation of a detector like this. This work, as well as the background reduction efforts that we advocate for here, is ongoing within the CYGNO collaboration [142]. Our results suggest a promising future for this technology. The next stage of CYGNO and the detectors of other groups within the Cygnus consortium [101] will be the first in a sequence of gas-based directional detectors to bring a range of complementary capabilities to the fleet of worldwide neutrino observatories.

ACKNOWLEDGMENTS

We thank the members of the CYGNO and Cygnus collaborations for discussions, and in particular Elisabetta Baracchini and Samuele Torelli for providing details about CYGNO's simulation results. CL and CAJO are supported by the Australian Research Council under the grant numbers DE220100225 and CE200100008. LES, NM, ACS are supported by the DOE Grant No. DE-SC0010813.

Appendix A: Effective neutrino oscillations

In Sec. II C 1, when describing the computation of the density matrix in the context of NSIs, we referred to the effective description of the neutrino oscillations in the Sun within the OMD approximation, as well as the averaging over the unknown neutrino production location. In this Appendix, we give more details of this calculation.

Within the OMD approximation, the Hamiltonian in Eq.(22) can be rotated by O ≡ R23R13, such that the rotated Hamiltonian H˜ = O†HO, and its evolution S˜ prop = O†SpropO, is given by,

$$\tilde{H} = \begin{pmatrix} H^{(2)} & 0 \ 0 & \Delta{33} \end{pmatrix}, \quad \tilde{S}{prop} = \begin{pmatrix} S^{(2)} & 0 \ 0 & e^{-i\Delta_{33}L} \end{pmatrix} \; . \quad (A1) \label{eq:hamiltonian}$$

Here, ^H(2) is the effective 2×2 Hamiltonian and ^S (2) ^≡ Evol[H(2)], describes the evolution in an effective 2 flavour scenario. Following the conventions in Refs. [46, 48, 71, 146] we get,

$$\begin{split} H^{(2)} &= \frac{\Delta m{21}^2}{4E\nu} \begin{pmatrix} -\cos 2\theta{12} & \sin 2\theta{12} e^{i\delta{\mathrm{CP}}} \ \sin 2\theta{12} e^{-i\delta{\mathrm{CP}}} & \cos 2\theta{12} \end{pmatrix} \ &+ \sqrt{2} G_F Ne(x) \begin{pmatrix} c{13}^2 - \epsilon_D & \epsilon_N \ \epsilon_N & \epsilon_D \end{pmatrix} \,, \end{split} \tag{A2}$$

where the terms ϵ^D and ϵ^N are related to the combined NSI parameter ϵαβ defined in Eq.(23) by,

$$\epsilon{D} = c{13}s{13}(s{23}\epsilon{e\mu} + c{23}\epsilon{e\tau}) - (1 + s{13}^{2})c{23}s{23}\epsilon_{\mu\tau}

• \frac{1}{2}c{13}^{2}(\epsilon{ee} - \epsilon{\mu\mu}) + \frac{1}{2}(s{23}^{2} - (s{13}c{23})^{2})(\epsilon{\tau\tau} - \epsilon{\mu\mu}), \epsilon{N} = c{13}(c{23}\epsilon{e\mu} - s{23}\epsilon{e\tau}) + s{13}[(s{23}^{2} - c{23}^{2})\epsilon{\mu\tau}
• c{23}s{23}(\epsilon{\tau\tau} - \epsilon{\mu\mu})].$$ (A3)

We then define the effective matter potential ACC and the matter-modified mixing p and q,

$$A_{\rm CC} = 2\sqrt{2}G_F Ne(x)E\nu \,, \tag{A4}$$

$$p = \left| \sin(2\theta{12})e^{i\delta{\text{CP}}} + \frac{2A_{\text{CC}}\epsilonn}{\Delta m{21}^2} \right|, \tag{A5}$$

$$q = \cos(2\theta{12}) - \frac{2A{\rm CC}}{\Delta m{21}^2} \left(\frac{c{13}^2}{2} - \epsilon_d\right),$$ (A6)

to obtain the following relations for the mass-squared difference and mixing angle in matter:

$$\Delta m{21,m}^2 = \Delta m{21}^2 \sqrt{p^2 + q^2},$$ (A7)

$$\sin(2\theta_m) = \frac{p}{\sqrt{p^2 + q^2}}, \tag{A8}$$

$$\cos(2\theta_m) = \frac{q}{\sqrt{p^2 + q^2}}.$$ (A9)

Taking the spatial average of cos(2θm) over the different production zones within the Sun, weighted by the fraction of neutrinos produced in each zone f(x), we get,

$$\langle \cos(2\theta_m) \rangle = \int_0^1 \cos(2\theta_m(x)) f(x) \, \mathrm{d}x,$$ (A10)

where x = r/R^⊙ is the normalised solar radial coordinate. The production site distributions, f(x), are different for each flux, and are taken from Ref. [147].

Appendix B: Coherent elastic neutrino-nucleus scattering

1. Standard Model

Given a nucleus N of mass m^N , under the SM the CEνNS process νN → νN occurs only through neutral current interactions via the Z exchange, which couples to all neutrino flavours equally. As a result, this process is flavour-blind at tree-level, and the cross section is given by,

$$\left(\frac{\mathrm{d}\sigma}{\mathrm{d}E_r}\right)^{\mathrm{SM}} = \frac{G_F^2 m_N F^2(q) g_V^2}{\pi} \left(1 - \frac{m_N Er}{2E\nu^2}\right). \quad (B1)$$

For a given nucleus with atomic number Z, mass number A and the number of neutrons N = A − Z, the SM coupling g^V = Zg^{V p} + Ng^V n . Here, g V ^p = 2g V ^u + g V ^d = ^{2 −} 2 sin² θ^w and g V ⁿ = g V ^u + 2g V ^d = − ² which signify the couplings with protons and neutrons respectively, in terms of the u and d couplings. The form factor, F(q), is used to capture the effects of nuclear structure, which become important at large momentum transfer, q = √ 2m^N Er. The commonly-used Helm ansatz for the form factor is [148]:

$$F(q) = \frac{3j_1(qr_n)}{qr_n} \exp\left(\frac{-q^2s^2}{2}\right)$$ (B2)

where j1(q) is the spherical Bessel function of first kind, rⁿ = q 3 (R² min − 3s ²) where s = 0.9 fm is a measure of the nuclear skin thickness and Rmin is the root-meansquare radius of the proton distribution of the nucleus. The form factor plays an essentially negligible role for recoil energies and nuclear targets we are interested in here, so we do not require any more sophisticated models than this—see e.g. Ref. [149].

2. NSIs for neutrino-nucleus scattering

Considering again the pure vector NSI case, we now take the Lagrangian in Eq.(13) and set f = {u, d} to build the modified CEνNS process. Assuming that the new light mediator has the same coupling with both u and d quarks, the new interactions can be accounted for in the cross section in Eq.(B1) by modifying,

$$g_V^{\rm NSI} = gV^{\rm SM} + \frac{3g{Z'}^2 Q{Z'}^f Q{Z'}^{\nu_\alpha} (Z+N)}{\sqrt{2} G_F (2m_N Er + m{Z'}^2)} \,. \eqno(B3)$$

For a heavy mediator, mZ′ ≫ q, we consider the Lagrangian in Eq.(14) with f = {u, d}. Following the same procedure as described in the main text in the context of EνeS, the generalised cross section for CEνNS can be written as [46],

$$\left(\frac{\mathrm{d}\zeta}{\mathrm{d}Er}\right){\alpha\beta}^{\mathrm{NSI}} = \frac{G_F^2 m_N F^2(q)}{\pi} \left(1 - \frac{m_N Er}{2E\nu^2}\right) \times \left[gV^2 \delta{\alpha\beta} + 2gV G{\alpha\beta} + \sumf G{\alpha f} G_{f\beta}\right], \tag{B4}$$

where,

$$G{\alpha\beta} = Z(2\varepsilon{\alpha\beta}^u + \varepsilon{\alpha\beta}^d) + N(\varepsilon{\alpha\beta}^u + 2\varepsilon_{\alpha\beta}^d)$$ (B5)

The double-differential event rate for CEνNS can also be written in exactly the same way as for EνeS cf. Eq.(36)—up to replacing the electron mass m^e by the nucleus mass m^N , and using N^T to count the number of nuclei in the target.

3. Nuclear recoils in directional gas detectors

The directions and energies of nuclear recoils are also independently measurable in a gas detector through essentially the same procedure as with electron recoils. In the case of nuclear recoils, however, the sizes of the tracks at the relevant energies (1 −15 keV^r for the nuclei in our chosen gas mixture) are much shorter, at ∼cm in length. This makes the effects of diffusion as the charge is transported to the readout plane a major obstacle for achieving good directionality. While directionality can still be used for background rejection, the near-disappearance of

FIG. 7. Projected median 90% CL exclusion limits for a directional gas detector at the required scale needed to measure CEνNS with solar neutrinos, i.e. a volume of 1000 m³ with atmospheric pressure gas. We assume a nuclear recoil threshold of 3 keV^r and performance as described in Appendix B 3. We project constraints onto the two-dimensional plane of up and down type NSI couplings across three example flavour-conserving interaction combinations, i.e. ee, µµ and τ τ . For comparison, we show the constraints from Ref. [48], for XENONnT's recent measurement of ⁸B neutrinos [23]. This represents a further future and less immediate goal for a potential large-scale recoil observatory, as designed for sensitivity to dark matter [101].

any details of the recoil tracks at low energies makes this challenging. This is coupled with the fact that we need to extend to much larger volumes to acquire sufficient statistics in the case of CEνNS, since the only detectable flux, ⁸B, is significantly weaker than that of pp neutrinos. This is why we demote this discussion to an appendix, but nevertheless, since this channel may be of interest in the longer term, we show some example results here.

Largely inspired by the extensive study of nuclear recoil directionality performed in the context of dark matter and solar CEνNS in Ref. [101], we adopt an ambitious but not unrealistic energy threshold of 3 keVr, taking all nuclear recoil energies to be those before quenching. We set the following performance parameters:

$$\left{ a{\rm HT}, \frac{E{\rm HT}}{\rm keV}, \frac{p_{\theta}}{\rm deg.}, p_E \right} = {0.1, 10.0, 25, 0.06} \quad (B6)$$

using the same functional forms for energy resolution, head-tail efficiency and angular resolution. Unlike in the electron recoil case, the relatively straight nature of the nuclear recoil tracks, combined with the washing-out of the ionisation distribution due to the diffusion, makes head-tail much more challenging to measure.

In Fig. 7 we show forecasted 90% CL exclusion limits on the two-dimensional space of u and d-type couplings, i.e. ε u αα, ε d αα across several possible flavour-conserving combinations. Results for other parameters are qualitatively similar. As can be readily observed from the plots, an ambitious volume of 1000 m³ and an aggressive background reduction in the nuclear recoil channel down to 50 times the ⁸B neutrino rate is required to obtain a level of sensitivity already met by XENONnT and PandaX via their detection of this solar flux. This conclusion is in general alignment with the sensitivity projections made in Ref. [101]. Although detecting solar CEνNS using this kind of experiment may be somewhat impractical in the near term, we expect the situation to be much more promising for detecting CEνNS with a higher-energy neutrino source, which we will explore in a future study.

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