WIMP Recoil Rates and Exclusion Plots Brian, April 2007 Section 1. Topics discussed in this note 1. Event rate calculation. What are the total and differential recoil rates from WIMPs with mass m χ and nucleon-normalized cross-section σ WN? More precisely, we need rates in terms of nucleon-normalized cross-sections σ SI, σ SD(p), and σ SD(n), the cross-sections for spin-independent, spin-dependent proton, and spin-dependent neutron couplings. 2. Cross-section vs. mass exclusion plots. We will extract from our data an event rate R above an energy threshold E T, which we claim with some statistical confidence is less than the rate from WIMPs. How do we generate exclusion plots from that rate, i.e. what is σ WN (R, E T, m χ )? 3. a n -a p exclusion plots. Related to the previous question, how do we generate spindependent exclusion plots in the a n -a p plane? Section 2. Additional files and references References are given in Section 8 Most importantly, see Lewin and Smith, for a good discussion on rate calculations. This is where I got most of my equations. Code files can be downloaded from kicp.uchicago.edu/~odom/analysis/rates 1. WIMP_rates_code.nb (main Mathematica file for calculations) 2. WIMP_rates_code.pdf (pdf version of previous) 3. WIMP_rate_checks_code.nb (various consistency checks, Mathematica file) 4. WIMP_rate_checks_code.pdf (pdf version of previous) Section 3. Halo density and velocities In the plots shown here, I use halo constants from a recent paper (Savage). These values are: characteristic WIMP velocity through the halo v 0 = 220 km/s, escape velocity v esc = 650 km/s, average earth velocity v E = 232 km/s, and halo density ρ = 0.3 GeV/cm 3. These parameters are consistent with those generally used in the literature. We can easily modify all calculations if someone has different favorite parameters they prefer to use. We will of course document our choice in a publication. Section 4. Event rate calculation The calculation is fairly straightforward for spin-independent interactions (see Lewin and Smith). I use the standard Maxwellian distribution with a cutoff for escape velocity. For spin-dependent interactions, the spin-composition and form factors are a little more difficult to handle (see Section 7.)
Checks: 1. My differential event rate vs. energy threshold calculation matches Juan s independent calculation, for spin-independent scattering on iodine. ctsêkevêkgêday 0.05 0.04 0.03 0.02 0.01 5 10 15 20 25 30 energy HkeVL Figure: Comparison between Juan s and my iodine SI differential event rate, for m χ = 100 GeV and σ SI = 10-6 pb and for m χ = 100 GeV and σ SI = 10-6 pb, using Juan s slightly different halo parameters. Dots: Juan s calculation, using the Gaussian scatterer form factor, Lewin and Smith Eq. (4.4). Red: Attempt to duplicate Juan s result with my machinery. Blue: My calculation, using the Helm FF, Lewin and Smith Eq. (4.7).
2. Juan and Makoto Minowa independently produced matching calculations for the SD(p) differential rate vs. recoil energy, using the odd-group spin model (see Section 7 for spin model discussion). Makoto has produeced SD dark matter limits from LiF bolometers. Figure: Differential scattering rate of WIMP SD(p) interactions with fluorine, where Juan s and Makoto Minowa s independent calculations are shown to produce identical results. The halo model used here is just slightly different from that used in the rest of this note.
3. My calculation for differential rate vs. recoil energy matches Juan s independent calculation, for spin-dependent-proton scattering on fluorine. ctsêkevêkgêday 1200 1000 800 600 400 200 5 10 15 20 25 30 energy HkeVL Figure: Comparison of Juan's and my fluorine SD(p) differential event rates for m χ = 10 GeV, σ SD(p) = 10 pb, and for m χ = 100 GeV, σ SD(p) = 110 pb. Dots: Juan's calculation using the odd-group spin model. Red: attempt to exactly duplicate Juan's result with my machinery. Blue: My more accurate calculation, using Pacheco s spin model, but still using Juan's infinite escape velocity. Green: My full calculation, still using Pacheco values. The effect of these model differences on inferred crosssections is shown in WIMP_rate_checks.pdf. Section 5. Cross-section vs. mass exclusion plots To make a plot of nucleon-normalized cross-section vs. mass, the expression R(σ WN, m χ, E T ) is inverted to obtain σ WN (R, m χ, E T ). For composite targets such as CF 3 I, the fractional mass of each target type must be taken into account when calculating an overall event rate per target mass. The nucleon-wimp cross-section follows the formula 1/σ WN = (1/σ WN(A1) + 1/σ WN(A2) +... ), where σ WN(An) is the cross-section inferred from
each type of nucleus in the target. This procedure is followed for each channel of interaction to obtain σ SI, σ SD(p), and σ SD(n). SI cross section HpbL 10 0.1 0.001 0.00001 5 10 20 50 WIMP mass HGeVL Figure: The black curve shows the inferred nucleon-normalized SI cross section σ SI as a function of WIMP mass, from an overall event rate R = 1 ct/kg/day and E T = 5 kev on a CF 3 I target. The cross-sections (σ WN(An) ) for each type of nucleus are shown for carbon (green), fluorine (red), and iodine (blue). In our case, there are a few ways we might imagine creating exclusion plots. In the simplest case, we start with a total event rate which we claim with 95% confidence to be greater than the actual WIMP event rate. If you take that rate and follow the above procedure, you will produce a curve corresponding to the 95%-excluded rate. All crosssections above that curve are 95% excluded. Checks: 1. Juan and I have generated consistent cross-section vs. mass curves, given a specified event rate and energy threshold. 2. The inversion of R(σ WN, m χ, E T ) to σ WN (R, m χ, E T ) is simple, so once we are confident of the integrated event rate calculations, checking the inversion by hand at a few points is all that is required. 3. For non-threshold experiments, conversion of spectra to exclusion plots is not always transparent, so direct comparison can be difficult. But, the exclusion plots we generate are clearly consistent with those generated by other experiments, scaled by rate and target nucleus.
Section 6. a n -a p exclusion plots The SD WIMP-nucleus cross-section is proportional to (a n <S n > + a p <S p >) 2, where a n and a p are WIMP-proton and WIMP-neutron coupling constants of unknown magnitude and sign, and <S n > and <S p > are the proton and neutron spin expectation values for the nucleus. There is a one-to-one correspondence between a n and σ SD(n) and between a p and σ SD(p). The proton and neutron terms interfere with each other, making useful an additional type of exclusion plot, where allowed regions are ellipses in the a n -a p plane. Each of these plots is only for a single WIMP mass. To generate this exclusion plot, all you need is the σ SD(p) limit and the σ SD(n) limit for the mass of interest, along with the sign of the ratio <S n > / <S p >. You need these quantities for each element of the target. To make the above plot, I used a ruler to read off values from published cross-section vs. mass exclusion plots, from the various experiments. Checks: The a n -a p plots I generate look identical to other people s published plots. I checked the intersection points with of the ellipses with certain horizontal or vertical lines in the plane, and found good agreement with various publications. See, for example, Tovey, Giuliani, and Savage.
Section 7. Form factors and nuclear spin model 1. General information See Lewin and Smith for a discussion. For SI calculations, I use the Helm form factor. For SD calculations, there are three ways to calculate WIMP coupling to the nuclear spin, corresponding to three models of nuclear spin structure: a) assume coupling to only unpaired nucleons, e.g. a single nucleon for a spin-1/2 nucleus (nobody actually cuts corners this far), b) only calculate coupling to the odd-group, e.g. coupling to all the protons in the case of fluorine, and c) do a full calculation of coupling to all nucleons. In the full calculation, there are three spin terms (proton, neutron, and interference), and each of them has their own form factor. A generic form factor exists for SD interactions (see Lewin and Smith), but as mentioned above, full spin model calculations yield more specific form factors. Carbon has strictly zero spin-dependent coupling because of conservation of angular momentum and its zero nuclear spin. Full spin model calculations are available for both fluorine and iodine. For fluorine, there are two choices: Pacheco and Divari. Pacheco does not actually have the individually calculated form factor for each of the three terms, but most people (see e.g. Tovey, Giuliani) seem to use his numbers, and they gives much better limits for SD(n). Pacheco s lack of specific form factors does not affect the exclusion plot limits significantly.
2. How important are form factors and choice of spin model? a. For COUPP, SI form factors are important for iodine, for WIMP masses above 10 GeV. SI form factors for fluorine and carbon are not important. SI cross section HpbL 10 0.1 0.001 0.00001 5 10 20 50 100 WIMP mass HGeVL Figure: Effects of form factor on inferred SI cross-section for E T = 5 kev and R = 1 ct/kg/day. Carbon curves are shown for a flat FF and for the Helm FF (both green, overlapping). Fluorine curves, overlapping one another, are shown for a flat FF (dashed dk. red) and for the Helm FF (red). Iodine curves are shown for a flat FF (dashed dk. blue) and for the Helm FF (blue) b. For COUPP, choice of spin model does not have a large effect on the SD(p) exclusion plot, but it does make a significant difference for the SD(n) exclusion plot.
SDHpL cross section HpbL 10 5 1 0.5 0.1 0.05 WIMP mass HGeVL 5 10 50 100 500 1000 Figure (above): Effects of form factor and spin model on inferred SD(p) crosssection for E T = 5 kev and R = 1 ct/kg/day. Fluorine curves are shown for a flat FF (with q=0 matching the Pacheco values) (dashed dk. red), Pacheco spin structure (red), and for Divari s spin structure (orange). Iodine curves are shown for a flat FF (dashed dk. blue), generic SD FF (aqua), and a full spin model (blue). SDHnL cross section HpbL 1000 100 10 1 WIMP mass HGeVL 1 5 10 50 100 500 1000 Figure (above): Effects of form factor and spin model on inferred SD(n) crosssection for E T = 5 kev and R = 1 ct/kg/day. Fluorine curves are shown for a flat FF (with q=0 matching Pacheco values) (dashed dk. red), Pacheco s spin structure
(red), and for Divari s spin structure (orange). Iodine curves are shown for a flat FF (dashed dk. blue), generic SD FF (aqua), and a full spin model (blue). 3. Checks a. Plots of SI form factors for iodine over the relevant range match those in the literature (see WIMP_rate_checks.pdf). b. For fluorine, my code shows that two completely different models yield identical results over the range of interest (see WIMP_rate_checks.pdf). Calculations for iodine, although the FF here is not important to us, exhibit reasonable agreement between a generic SD form factor and the full iodine calculation. 4. Conclusions The choice of SI form factor is straightforward. For iodine spin composition and SD form factor, we should use Ressel s results, although this only affects us insofar as our a n -a p ellipses get very slightly more or less long depending on which form factor is used. For the fluorine spin composition and SD form factor, I currently have emails out asking why people have chosen Pacheco s spin structure (the one which makes our SDn results look better) over Divari s. I expect that we will end up following suit and using Pacheco s spin structure when we construct our exclusion plots. Section 8: References 1. Lewin and Smith, Astroparticle Physics 6 (1996) 87. (general overview) 2. Savage, Gondolo, and Freese. Phys Rev D 70 (2004) 123513 (halo model) 3. Tovey, et.al, Phys Lett B 488 (2000) 17-26. (an-ap exclusion plots) 4. Giuliani, Phys Rev Lett 93 (2004) 161301 (an-ap exclusion plots) 5. Pacheco and Strottman, Phys Rev D 40 (1989) 2131 (fluorine spin structure) 6. Divari, et. al, Phys Rev C 61 (2000) 054612 (fluorine spin structure) 7. Ressel and Dean, Phys Rev C 56 (1997) 535 (iodine spin structure)