LUX Formalism in Dark Matter Experiments

• LUX Formalism is a calibrated framework that translates dark-matter interaction models into xenon-TPC observables using S1/S2 signal measurements.
• It employs profile-likelihood inference with nuisance-parameter constraints to derive exclusion limits under explicit astrophysical and detector-response assumptions.
• The formalism extends to sub-GeV dark matter by modeling Bremsstrahlung and Migdal effects, converting minimal nuclear recoils into detectable electron-recoil signals.

Searching arXiv for the specified LUX papers and closely related formalism papers. The LUX formalism is the integrated analysis framework used by the Large Underground Xenon experiment to map dark-matter interaction models into observable distributions in a dual-phase liquid-xenon time projection chamber, calibrate those observables with in situ electron-recoil and nuclear-recoil data, and derive exclusion limits through profile-likelihood inference under explicit astrophysical and detector-response assumptions. In the LUX literature, the formalism appears in several closely related forms: a canonical elastic WIMP-search pipeline for spin-independent and spin-dependent scattering [(Collaboration et al., 2013); (Collaboration et al., 2015); (Silva, 2017)], a nuclear-recoil calibration and microphysics framework anchored by D–D neutron scattering (Collaboration et al., 2016), and an extension to sub-GeV dark matter using Bremsstrahlung and Migdal electron-recoil signatures when the nuclear recoil itself is below threshold (Akerib et al., 2018).

1. Detector basis and observable space

LUX is a dual-phase xenon TPC in which an interaction in liquid xenon produces prompt scintillation, denoted S1, and ionization electrons that are drifted upward and extracted into the gas to generate proportional scintillation, denoted S2 (Silva, 2017). The detector records both signals with 122 PMTs arranged in top and bottom arrays, and reconstructs depth from the S1–S2 drift time while the transverse position is inferred from the S2 light pattern on the top array (Silva, 2017). In the first-results configuration, the active liquid xenon target was 250 kg within a TPC of 47 cm diameter and 48 cm drift height, with photodetection via two arrays of 61 PMTs (Collaboration et al., 2013).

A central feature of the LUX formalism is that analysis is performed directly in detector observables rather than in recoil energy alone. In the early 2013 analysis, the unbinned extended profile-likelihood ratio used four observables, radius, depth, S1, and S2$_b$, where S2$_b$ denotes S2 measured from the bottom PMT array alone (Collaboration et al., 2013). In the 2013 reanalysis, the signal and background PDFs were constructed in $(cS1, cS2, r, z)$, where weekly $^{83\mathrm{m}}$Kr calibrations were used to define corrected observables that equalize detector response throughout the active volume (Collaboration et al., 2015). In the combined WS2013 and WS2014–16 analysis, the observable spaces were $\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$ for WS2013 and $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$ for WS2014–16, reflecting the more complicated field geometry of the later run (Silva, 2017).

The observables are expressed in units of detected photons (phd), not photoelectrons, to account for double photoelectron emission (Silva, 2017). LUX also used photon counting for low S1 to improve resolution (Silva, 2017). This detector-level representation is foundational: it is the space in which both calibration data and dark-matter signal models are compared to the search sample.

2. Response model, gains, and energy reconstruction

The LUX formalism connects deposited energy to measured signals through a quanta model. For electron recoils, the reconstructed energy is obtained by combining scintillation and ionization according to

$$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$

with $W = 13.7 \pm 0.2$ eV per quanta (Akerib et al., 2018). For the WS2013 sub-GeV analysis, the gains were $g_1 = 0.117$ phd/photon and $g_2 = 12.2$ phd/electron, with electron extraction efficiency $_b$0 (Akerib et al., 2018). The combined WIMP-search formalism reports closely related WS2013 values $_b$1 phd/photon and $_b$2 phd/electron (Silva, 2017), while the 2013 reanalysis quotes $_b$3 phd per scintillation photon and $_b$4 phd per extracted electron, with anticorrelation $_b$5 (Collaboration et al., 2015).

For nuclear recoils, the response model is expressed through the light yield $_b$6 and charge yield $_b$7, which determine the number of photons and electrons: $_b$8 and then

$_b$9

(Collaboration et al., 2016). The D–D calibration paper gives $(cS1, cS2, r, z)$0 phd/photon and $(cS1, cS2, r, z)$1 phd/electron for the calibration at $(cS1, cS2, r, z)$2 V/cm, with extraction efficiency $(cS1, cS2, r, z)$3 (Collaboration et al., 2016).

The response model in later analyses was implemented with NEST, tuned to calibration data. In WS2014–16, variable electric fields necessitated splitting the exposure into 16 subdatasets, each with a unique NEST response model constrained by tritiated methane and D–D neutron calibrations (Silva, 2017). The proceedings summary states that LUX used NEST to generate NR and ER Monte Carlo signal and background models for each of the 16 time/z bins, with yield formulae updated using LUX’s own tritium and D–D calibrations (Szydagis, 2016).

A plausible implication is that “LUX formalism” denotes not one equation set but a calibrated forward model: quanta production, transport, gain, correction, and fluctuation modeling are all embedded before any likelihood evaluation.

3. Nuclear-recoil microphysics and in situ calibration

The D–D neutron calibration established the nuclear-recoil branch of the LUX formalism by reconstructing recoil energy from double-scatter neutron kinematics (Collaboration et al., 2016). The analysis defines the recoil energy using the center-of-mass scattering angle and employs the approximation

$(cS1, cS2, r, z)$4

with better than $(cS1, cS2, r, z)$5 accuracy for all angles (Collaboration et al., 2016). This kinematic reconstruction enabled absolute in situ measurements of $(cS1, cS2, r, z)$6 from $(cS1, cS2, r, z)$7 to $(cS1, cS2, r, z)$8 keV$(cS1, cS2, r, z)$9 plus an endpoint at $^{83\mathrm{m}}$0 keV$^{83\mathrm{m}}$1, and $^{83\mathrm{m}}$2 from $^{83\mathrm{m}}$3 to $^{83\mathrm{m}}$4 keV$^{83\mathrm{m}}$5 plus an endpoint at $^{83\mathrm{m}}$6 keV$^{83\mathrm{m}}$7 (Collaboration et al., 2016).

The calibration demonstrated measurable signals down to $^{83\mathrm{m}}$8, with representative values near $^{83\mathrm{m}}$9 keV$\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$0 of $\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$1 and $\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$2 (Collaboration et al., 2016). At $\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$3 keV$\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$4, the reported representative values are $\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$5 and $\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$6 (Collaboration et al., 2016). The paper states that these results extend measurable NR response down to $\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$7 keV$\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$8 and improved low-mass WIMP sensitivity by a factor of $\{r_{\mathrm{ver}}, z_{\mathrm{ver}}, S1, S2\}$9 at $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$0, reducing the lowest accessible WIMP mass from $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$1 to $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$2 (Collaboration et al., 2016).

The microphysical energy scale uses the $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$3-value and an electronic energy fraction $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$4: $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$5 (Collaboration et al., 2016). LUX fit both a Lindhard-based model and a Bezrukov/Ziegler-based parameterization to the data, with a Lindhard best-fit parameter $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$6 and a biexcitonic quenching parameter $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$7 (Collaboration et al., 2016). In the 2013 reanalysis, the mean fraction of energy going to quanta for NRs was parameterized by the Lindhard model with $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$8, treated as a Gaussian-constrained nuisance parameter in the likelihood (Collaboration et al., 2015).

These calibrations replaced earlier conservative assumptions. The 2013 reanalysis explicitly states that the previous analysis modeled the signal only above $\{r_{S2}, \phi_{S2}, \tau_d, S1, S2\}$9 keV minimum energy, whereas the revised analysis truncated the WIMP signal only below the lowest D–D calibration point of $$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$0 keV (Collaboration et al., 2015). This change dominates the low-mass improvement (Collaboration et al., 2015).

4. Astrophysical and scattering-rate framework

For elastic scattering, LUX adopts the standard differential-rate formalism under the Standard Halo Model. The 2013 first-results paper writes

$$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$1

with

$$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$2

where $$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$3 is the Helm form factor, and

$$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$4

for isospin-invariant spin-independent coupling (Collaboration et al., 2013). The minimum speed is

$$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$5

(Collaboration et al., 2013).

The halo assumptions are stated explicitly in the early WIMP-search literature. The first-results paper uses

$$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$6

(Collaboration et al., 2013). The reanalysis preserves the same Standard Halo Model structure and again lists $$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$7, $$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$8, $$E = W\left(n_{\gamma}+n_e\right)=W\left(\frac{S1}{g_1}+\frac{S2}{g_2}\right),$$9, and $W = 13.7 \pm 0.2$0 (Collaboration et al., 2015). The combined WIMP-search paper states that LUX adopts the same Standard Halo Model assumptions used by XENON and CDMS, with $W = 13.7 \pm 0.2$1 for WS2013 and $W = 13.7 \pm 0.2$2 for WS2014–16, while other SHM parameters are standard in the cited works but not explicitly listed there (Silva, 2017).

For spin-dependent scattering, the combined analysis summarizes the common LUX SD formalism in terms of nuclear structure functions $W = 13.7 \pm 0.2$3: $W = 13.7 \pm 0.2$4 (Silva, 2017). The same source notes that neutron-only and proton-only limits are quoted as $W = 13.7 \pm 0.2$5 and $W = 13.7 \pm 0.2$6 (Silva, 2017).

The proceedings summary presents the scattering-rate formalism in a compact form and emphasizes that LUX combined standard astrophysical assumptions, isospin-invariant couplings, and in situ detector calibrations in the same pipeline (Szydagis, 2016). This suggests that the formalism’s defining feature is not any single interaction model, but rather the reproducible passage from halo model and cross section to observable-space PDFs.

5. Profile-likelihood inference, backgrounds, and datasets

The statistical core of the LUX formalism is an extended profile-likelihood ratio analysis with nuisance-parameter constraints. In the first-results paper, the extended likelihood for $W = 13.7 \pm 0.2$7 observed events with signal mean $W = 13.7 \pm 0.2$8 and background means $W = 13.7 \pm 0.2$9 is written schematically as

$g_1 = 0.117$0

(Collaboration et al., 2013). One-sided $g_1 = 0.117$1 confidence limits were then set on the SI WIMP–nucleon cross section by profiling the likelihood over nuisance parameters (Collaboration et al., 2013).

The 2013 reanalysis likewise used a double-sided profile-likelihood ratio. For each WIMP mass, an unbinned extended likelihood over $g_1 = 0.117$2 was built as a mixture of signal and background PDFs with Gaussian-constrained nuisance parameters, including the NR-response parameters $g_1 = 0.117$3 and $g_1 = 0.117$4, as well as background normalizations (Collaboration et al., 2015). Monte Carlo pseudo-experiments via RooStats were used to construct $g_1 = 0.117$5 confidence intervals, and a power constraint at the median expected limit was applied to avoid over-exclusion due to downward background fluctuations (Collaboration et al., 2015).

The combined WS2013 and WS2014–16 analysis used an unbinned PLR across 17 exposure segments, with WS2013 as a 17th segment and WS2014–16 divided into 16 time–drift bins (Silva, 2017). The generic likelihood form is given as

$g_1 = 0.117$6

(Silva, 2017).

The background model is a major part of the formalism. In the 2013 reanalysis, the ER background model included gamma rays, beta decays, $g_1 = 0.117$7Xe, $g_1 = 0.117$8Ar, and a new empirical model for wall-originating events that enabled an enlarged fiducial radius to 20 cm (Collaboration et al., 2015). The fitted background normalizations are reported explicitly, including $g_1 = 0.117$9, $g_2 = 12.2$0, $g_2 = 12.2$1, $g_2 = 12.2$2, $g_2 = 12.2$3, and $g_2 = 12.2$4 (Collaboration et al., 2015). In the sub-GeV analysis, six background components were constrained with Gaussian priors: low-z gamma rays, other gamma rays, beta particles, $g_2 = 12.2$5Xe, $g_2 = 12.2$6Ar, and wall events (Akerib et al., 2018).

Several concrete datasets define the canonical LUX implementations. The first-results search used $g_2 = 12.2$7 live days with a fiducial mass $g_2 = 12.2$8 kg and yielded 160 events after cuts in the ROI (Collaboration et al., 2013). The 2013 reanalysis used $g_2 = 12.2$9 live days, fiducial mass $_b$00 kg, and 591 events after cuts (Collaboration et al., 2015). The combined WIMP-search paper adds the WS2014–16 run with 332 live days and 1,221 events after cuts, for a total WS2014–16 exposure of $_b$01 kg·live-years (Silva, 2017).

6. Extension to sub-GeV dark matter: Bremsstrahlung and Migdal channels

The sub-GeV extension of the LUX formalism addresses a regime in which ordinary elastic nuclear recoils are often invisible in xenon because the deposited NR energy is below threshold (Akerib et al., 2018). The key observation is that the tree-level DM–nucleus scattering diagram can be accompanied by Bremsstrahlung photon emission or by the Migdal effect, producing an electron-recoil component at higher observable energy than the NR alone (Akerib et al., 2018).

LUX modeled these channels by starting from the standard SI DM–nucleus scattering framework and weighting the differential NR cross section by the probability of an accompanying ER process (Akerib et al., 2018). The schematic rate is given as

$_b$02

(Akerib et al., 2018).

For Bremsstrahlung, the calculation follows Kouvaris and Pradler, with

$_b$03

in a soft-photon approximation (Akerib et al., 2018). For the Migdal effect, LUX followed Ibe et al. and included only ionization, omitting excitation and excluding valence-electron shells $_b$04 because liquid-phase effects may perturb valence spectra (Akerib et al., 2018). The Migdal observable energy is

$_b$05

and the ionization probability density is summed shell by shell (Akerib et al., 2018).

This extension also introduced explicit mediator classes. Four mediator classes were tested: heavy or light, each with scalar or vector couplings (Akerib et al., 2018). Scalar mediator couplings scale coherently as $_b$06, while vector mediator couplings scale as $_b$07 (Akerib et al., 2018). The mediator form factor is parameterized as

$_b$08

and

$_b$09

(Akerib et al., 2018).

The experimental motivation is directly tied to thresholds. LUX had $_b$10 detection efficiency at $_b$11 keV for ERs versus $_b$12 keV for NRs, so ER-side channels opened sensitivity to $_b$13 below about $_b$14 where NR-only searches are inefficient (Akerib et al., 2018). The sub-GeV search used the 2013 WS2013 dataset with $_b$15 live days and search exposure $_b$16 kg·day, and set constraints on SI DM–nucleon scattering for masses $_b$17–$_b$18 (Akerib et al., 2018). The observed event counts were consistent with the background-only hypothesis for all tested masses and mediator classes (Akerib et al., 2018).

A plausible implication is that this sub-GeV analysis did not replace the canonical LUX formalism; it reused the same detector-response and PLR machinery while substituting ER-band signal models for otherwise invisible NR interactions.

7. Results, scope, and interpretive boundaries

Across its main WIMP analyses, the LUX formalism yielded successively stronger $_b$19 C.L. limits under standard halo assumptions. The first-results paper reported a minimum SI upper limit of $_b$20 at $_b$21 (Collaboration et al., 2013). The 2013 reanalysis improved this to $_b$22 zb, explicitly $_b$23, again at $_b$24 (Collaboration et al., 2015). The combined WS2013+WS2014–16 analysis reported a minimum SI limit of $_b$25, equivalently $_b$26, at $_b$27, and spin-dependent minima of $_b$28 for neutron-only coupling and $_b$29 for proton-only coupling at $_b$30 (Silva, 2017). The sub-GeV search extended sensitivity to $_b$31–$_b$32 using Bremsstrahlung and Migdal ER signatures (Akerib et al., 2018).

The formalism’s scope is defined by explicit assumptions and limitations. LUX’s principal WIMP analyses adopted the Standard Halo Model and did not profile astrophysical uncertainties in the first-results paper (Collaboration et al., 2013). The reanalysis states that uncertainties in the NR-response nuisance parameters changed the limit by less than $_b$33 relative to fixing the best-fit model (Collaboration et al., 2015). The sub-GeV paper notes that variations in $_b$34, $_b$35, and $_b$36 can shift limits modestly, that light-mediator definitions can differ slightly between experiments, and that omitting valence-shell Migdal contributions makes the limits conservative (Akerib et al., 2018).

The literature also contains reinterpretive analyses that stress the dependence of low-mass exclusions on low-energy liquid-xenon response assumptions. “Dark Matter in Light of LUX” studies exothermic scattering, isospin-dependent couplings, halo-independent comparisons, and variations in LXe scintillation assumptions, concluding that only a highly tuned isospin-dependent scenario remains viable unless LXe scintillation properties are dramatically different from those assumed by LUX (Fox et al., 2013). This does not revise the collaboration’s formalism, but it identifies where sensitivity to modeling assumptions is concentrated.

Taken together, the LUX formalism is best understood as a detector-calibrated, likelihood-based translation layer between dark-matter theory and xenon-TPC observables. Its stable components are the S1/S2 signal model, in situ ER and NR calibration, nuisance-constrained PLR inference, and Standard Halo Model rate calculation. Its adaptable components are the interaction model and the choice of signal channel: elastic NRs for canonical WIMPs, or ER-band Bremsstrahlung and Migdal signatures for sub-GeV dark matter (Silva, 2017, Collaboration et al., 2016, Akerib et al., 2018).