Signal Models: Dark-Matter Rates

3.12 Poisson Limit Setting

3.12.2 Signal Models: Dark-Matter Rates

As in HVeV Run 2, the dark-matter models of interest were those of dark-matter- electron scattering (DMe), dark-photon absorption (DPA), and axion-like particle absorption (ALP). In this section, we will discuss the equations used to calculate the expected event rates from each model as a function of electron recoil energy (𝐸_𝑒𝑟).

The differential scattering rate used for DMe can be seen in Eq. 3.21. This rate was derived in [24] by considering the excitation of bound electrons in a static potential (the crystal structure) via scattering with dark matter from the local dark-matter halo.

𝑑𝑅_{𝐷 𝑀 𝑒} 𝑑ln𝐸_𝑒𝑟

= 𝜌_{𝐷 𝑀} 𝑚_{𝐷 𝑀}

1 2𝑚_𝑆𝑖

𝑐𝜎_𝑒𝛼 𝑚²

𝑒

𝜇²

𝐷 𝑀

𝐼_crystal(𝐸_𝑒𝑟;𝐹_{𝐷 𝑀}) (3.21) Each parameter utilized is described below for reference.

• 𝑅_{𝐷 𝑀 𝑒}is the rate of dark-matter-electron scattering in units of events/exposure.

• 𝐸_𝑒𝑟 is the recoil energy of the electron.

• 𝜌_{𝐷 𝑀} is the local mass density of dark matter.

• 𝑚_{𝐷 𝑀} is the mass of a single dark-matter particle.

• 𝑚_𝑆𝑖 is the mass of a silicon nucleus.

• 𝑐is the speed of light in a vacuum.

• 𝜎_𝑒 is the interaction cross section (the model parameter of interest for DMe).

• 𝛼is the fine structure constant.

• 𝑚_𝑒is the mass of an electron.

• 𝜇_{𝐷 𝑀} is the reduced mass of the𝑚_{𝐷 𝑀}-𝑚_𝑒 system.

• 𝐼_crystalis the (unitless) scattering integral for DM-electron scattering.

• 𝐹_{𝐷 𝑀}is the momentum-transfer form factor, which determines the dependence of scattering on exchanged momentum (𝑞). For Run 3, we consider𝐹_{𝐷 𝑀} =1 and𝐹_{𝐷 𝑀} =(𝛼𝑚_𝑒/𝑞)²for exchange of a heavy and light mediator, respectively.

As in Run 2, 𝐼_crystalwas calculated using the QEdark code presented in [24] (made publicly available by the authors). For Run 3, the values were calculated using the dark-matter halo parameters recommended in [48]. This change (away from the Run 2 halo parameters) was made as part of a broader attempt to make SuperCDMS results directly comparable to those of other collaborations. The SuperCDMS, SENSEI, and DAMIC collaborations have all agreed to make use of these parameters.

The Run 2 and Run 3 halo parameters are compared in Table 3.1.

Halo parameter Run 2 value Run 3 value units

Local DM density 0.3 0.3 GeV/cm³

Average Earth velocity w.r.t. DM halo 240 253.7 km/s Average DM velocity w.r.t. galactic frame 230 238 km/s

Galactic Escape Velocity 600 544 km/s

Table 3.1: Table of halo parameters used for HVeV Runs 2 and 3. Except for density, the Run 2 values were taken from Section 6 of [24]. The Run 2 density was taken from [49]. The Run 3 values were taken from [48]. The Run 3 average Earth velocity (w.r.t. DM halo) was calculated using the average DM velocity (w.r.t.

galactic frame), the solar peculiar velocity, and the Earth velocity (w.r.t the Sun) on March 9th.

The expected DPA rate for Run 3 was calculated using Eqs. 3.22 and 3.23. These equations were derived in [50] by modifying the equation for absorption of photons by electrons in a semiconductor. The equation was modified to account for the dark-photon number density, kinetic mixing (𝜀), and non-relativistic velocity.

𝑅_{𝐷 𝑃 𝐴} = 1 𝜌_𝑆𝑖

𝜌_{𝐷 𝑀} 𝑚_𝜈

𝜀²

eff

𝜎₁(𝜔 =𝑚_𝜈^𝑐

2

ℏ)

ℏ (3.22)

𝜀²

eff= 𝜀²𝑚²

𝜈𝑐⁴ 𝑚²_𝜈𝑐⁴−2𝑚_𝜈𝑐²𝜎₂+𝜎²

2 +𝜎²

1

(3.23)

• 𝑚_𝜈 is the dark-photon mass. The dark photon is wholly absorbed, so 𝐸_𝑒𝑟 = 𝑚_𝜈𝑐².

• 𝜀_effis the (unitless) effective kinetic-mixing parameter in the silicon medium.

• 𝜎₁(𝜔)and𝜎₂(𝜔)are the real and imaginary parts of the complex conductivity of silicon (in units of energy) as a function of frequency𝜔.

• ℏis the reduced Planck constant.

• 𝜀is the (unitless) kinetic-mixing parameter in vacuum.

The expected ALP-absorption rate for Run 3 was calculated using Eq. 3.24. This equation was also derived in [50] by modifying the photon-absorption equation. The equation was similarly modified to account for the ALP number density, axioelectric coupling (𝑔_{𝑎 𝑒}), and non-relativistic velocity.

𝑅_{𝐴 𝐿 𝑃} = 𝜌_{𝐷 𝑀} 𝜌_𝑆𝑖

3𝑔²

𝑎 𝑒𝑚_{𝐴 𝐿 𝑃} 16𝜋𝛼𝑚²_𝑒

𝜎₁(𝜔 =𝑚_{𝐴 𝐿 𝑃}^𝑐

2

ℏ)

ℏ (3.24)

Parameters that were not also used in previous equations are described below for reference:

• 𝑅_{𝐴 𝐿 𝑃} is the rate of ALP absorption in units of events/exposure.

• 𝑔_{𝑎 𝑒} is the axioelectric coupling factor.

• 𝑚_{𝐴 𝐿 𝑃}is the ALP mass. The ALP is wholly absorbed, so𝐸_𝑒𝑟 =𝑚_{𝐴 𝐿 𝑃}𝑐².

Eq. 3.24 is slightly different from what was used in HVeV Run 2. The Run 2 equation can be found by setting 𝜎₁ → ℏ𝑐𝜎_{𝑝 .𝑒 .}𝜌_𝑆𝑖, where𝜎_{𝑝 .𝑒 .} is the silicon photoelectric- absorption cross section in units of area/mass. The relationship between complex conductivity and photoelectric-absorption cross section (Eq. 3.25) can be used to

show that both equations are equivalent when the silicon index of refraction (𝑛(𝜔)) goes to unity.

𝜎₁(𝜔) =𝑛(𝜔)𝜎_{𝑝 .𝑒 .}(𝜔)𝜌_𝑆𝑖ℏ𝑐 (3.25) The silicon𝑛(𝐸 = ℏ𝜔) can be seen in Figure 3.43. We see that𝑛goes to unity for deposited energies above 30 eV.

Figure 3.43: The silicon index of refraction as a function of photon energy (𝐸 = ℏ𝜔).

Figure is courtesy of Matt Wilson using data from [51].

Run 2 used the𝜎_{𝑝 .𝑒 .} version of the Eq. 3.24 as a carry-over from the SuperCDMS Soudan dark-photon and ALP analysis [27]. The Soudan limit was set for ALP masses≥40 eV and was therefore unaffected by the difference. We believe (following [50]) that Eq. 3.24 is the more accurate choice and ideally would have been used for Run 2. When comparing the Run 2 and 3 results, we scaled the Run 2 ALP limit to account for this difference.

For Run 3, we used silicon𝜎₁(𝐸 = ℏ𝜔) values curated by Matt Wilson (see Figure 3.44). Above 3.2 eV, these values were taken from literature ([51], [52], and [53]).

Below 3.2 eV, the values were from measurements and a fitted model of indirect absorption [54]. The fitted model produced nominal, upper and lower results. We used the nominal result to calculate the Run 3 limits. We used the upper and lower results when estimating uncertainty on the limits.

Figure 3.44: The real part of the complex conductivity of silicon used for the Run 3 absorption limits. Figure is courtesy of Matt Wilson. The colors and legend indicate where each region’s data originated. The indirect absorption model is from [54].

The handbook of optical constants is [51]. The Henke et. al. paper is [52]. The XCOM data is from [53].

The silicon 𝑛 and 𝜎₁ values used for Run 3 (along with 𝜎_{𝑝 .𝑒 .} and all analogous values for germanium) can be easily loaded using a now publicly available software package4.