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User: As a text script expert, please help me to write a short text script with the topic " The Partial Rate Matrix for Dark Matter Scattering Benjamin Lillard1,∗ 1Institute for Fundamental Science and Department of Physics, Willamette Hall, University of Oregon, Eugene, OR 97401, U.S.A. (Dated: January 10, 2025) I present a highly efficient integration method for scattering calculations, and a “partial rate matrix”thatencodesthescatteringrateasafunctionoftheSO(3)orientationofthedetector.The search for low mass (e.g. sub-GeV) R gχ locity distributions; N particle models of the DM, dark matter (DM) is driving a search for detector ma- DM specifying the DM mass m and scattering form factor terials with lower energy thresholds, including semicon- χ F ; and N different models for the detector physics ductors [1–5], molecular targets [6–8], and a variety of DM fS (e.g. for different target materials or final states).Predicting the DM scat- standard approach, the rate integral must be evaluated tering rate in such experiments requires input from con- N N N N many times, which soon becomes pro- densed matter or physical chemistry, complicating what R gχ DM fS hibitively expensive [63].tegration” that factorizes the astrophysics, the DM par- Materialswithdirectionallydependentscatteringrates ticle physics, and the SM physics of the target material havethecapabilitytodistinguishtheDMsignalfromany by projecting the velocity distribution g (⃗v) and the SM χ Standard Model (SM) backgrounds, but at a computa- momentum form factor f2(⃗q) onto bases of orthogonalInner products take the form ρ (cid:90) R=N χ dEd3qd3vg (⃗v)V S(⃗q,E) (cid:90) Tm χ χ cell ⟨f|ϕ⟩= d3vf(⃗v)ϕ(⃗v), (7) (cid:18) q2 (cid:19)(cid:18) σ¯ F2 (q)(cid:19) ×δ E+ −⃗q·⃗v 0 DM , (1) (cid:90) 2m 8π2µ2 ⟨f|φ⟩= dEd3qf(⃗q,E)φ(⃗q,E).(8) χ χ where⃗qandE arethemomentumandenergytransferred For L2 normalized basis functions, ⟨ϕ|ϕ⟩ = ⟨φ|φ⟩ = 1, f Dro Mm vD elM ocitt yo dth ise trS ibM utip oa nr ,ti nc ole r; mg aχ li( z⃗v e) di ts oa (cid:82)n do 3vnr gela (⃗vti )v =ist 1ic ; |g χ⟩ and (cid:12) (cid:12)f S2(cid:11) are Hilbert space representations of the χ originalfunctions.TheDM–SMscatteringisrepresented ρ is the local DM mass density; µ = m m /(m + χ χ χ SM χ by a matrix M, with coefficients: m ) is the reduced mass of the SM–DM system, forS stM rucp ta ur rt eic fl ue nm cta ioss nm [4S 7M , ; 48a ]n wd hV icce hllS en(⃗q c, oω de) sis tht ehe prd oy pn ea rm tieic s Mφ ≡(cid:28) ϕ(⃗v)(cid:12) (cid:12) (cid:12) F D2 M(q) δ(cid:18) E+ q2 −⃗q·⃗v(cid:19)(cid:12) (cid:12) (cid:12)φ(⃗q,E)(cid:29) , ϕ (cid:12)4πm µ2 2m (cid:12) of the target material, with volume normalization factor χ χ χ (9) V .∆E s, and V cellS is replaced by 2πf s2(⃗q)δ(E−∆E s): I show that R(R) can be evaluated extremely quickly once put into the form: (cid:90) (cid:88) R=N ρ σ¯ d3qd3vg (⃗v)f2(⃗q) T χ 0 χ s R(R)=N ρ σ¯ (cid:88) ⟨ℓm|R|ℓm′⟩K(ℓ) , (11) s T χ 0 mm′ F2 (q) (cid:18) q2 (cid:19) ℓ,m,m′ × DM δ ∆E + −⃗q·⃗v .(2) 4πµ2m s 2m χ χ χ with −ℓ ≤ m,m′ ≤ ℓ.Here K(ℓ) [g ,f2] is the partial mm′ χ S rate matrix, defined as: See e.g. [7, 46–49] for details.K(ℓ) ≡ P min g f2 F2 , (12) mm′ 2µ2m ℓ v ℓm ℓm′ DM I project the functions of ⃗v and (⃗q,E) onto vector χ χ spaces spanned by complete and orthogonal basis func- where g (|⃗v|) = ⟨ℓm|g ⟩ and f2 (|⃗q|,E) = ⟨ℓm′|f2⟩ tions |ϕ⟩ and |φ⟩: ℓm χ ℓm′ S are the projections of g (⃗v) and f2(⃗q,E) onto spherical χ S |ϕ (⃗v)⟩=r(v)(v)Y (vˆ), (3) harmonic modes, and v min(q,E)=E/q+q/(2m χ).Mφ simpli- C. Partial Rate Matrix ϕ fies dramatically in the spherical harmonic basis, espe- ciallywhenF (⃗q)isisotropic(e.g.forspin-independent DM As the penultimate step in the calculation, Eq.With g χ(⃗v) provided in organized into a series of partial rate matrices K(ℓ) by the lab frame, and with F DM(⃗q)=F DM(q), the operator completing the sums over n,n′ and j: in Eq.∞ ℓ ℓ Returning the (nℓm) and (jn′ℓ′m′) indices to |ϕ⟩ and R(R)=N ρ σ¯ (cid:88) (cid:88) (cid:88) G(ℓ) (R)K(ℓ) .(9) simplifies to [65]: T χ 0 mm′ mm′ ℓ=0m=−ℓm′=−ℓ Mj nn ℓm′ℓ′m′ ≡δ ℓℓ′ δ mm′ · I n(ℓ ,) jn′(m χ,F DM), (13) Each combination of g χ, f S2, m χ and Fgenerates dif- (cid:90) ∞ F2 (q) ferent K(ℓ), each of which contains all of the informa- I n(ℓ ,) jn′ = dEqdq 2D mM µ2 r j(q n) ′(q,E) tion that might be extracted by rotating the detector 0 χ and measuring the rate.(14) separates vmin(q,E) ℓ v n the rate into contributions from fixed ℓ, R = (cid:80) ℓR(ℓ).States of different radial index n,n′ are mixed by I, and where P is the ℓth Legendre polynomial, and v (q,E) the rotation matrices G(ℓ) mix the m,m′ of fixed ℓ; but ℓ min is defined in the usual way, the different ℓ modes do not mix.(15) χ min q 2m thedetector form factormusthavesupportatthatsame χ ℓ.ensures that q = 0 is outside the integration region for benefit: how the partial rate R(ℓ) falls as a function of ℓ any finite v, so that I remains finite [66].gχ fS DM K R Tr Here ϑ represents any external parameters, e.g. the DM where T p( rq o,v j,) is the average integration time for a single mass m χ.(18)foronecom- bination of g , f2 and DM particle model; and T is when many versions of each f i and ϑ must be tested si- χ S Tr the evaluation time for the ℓmm′ sum in Eq.Toaccount χ DM (cid:90) 1 for the uncertainty in g χ, such an analysis could be per- x2dxh (x)h (x)=δλ′ δµ′ , (A1) formed on an ensemble of realistic g models, or in a λµ λ′µ′ λ µ χ 0 model-independentwaybyvaryingthecoefficientsof|g ⟩ χ (in the spirit of e.g. [53–59]).Each wavelet is largeanisotropicdirectdetectionexperimentsmeasuring piecewise-constant, R(R) with precision, the experimentally determined val-  ues of K m(ℓ m) ′ can be translated into measurements of the  +A λµ x 1 ≤x<x 2, DM velocity distribution, in what is sometimes referred h (x)= −B x <x≤x , (A2) λµ λµ 2 3 to as DM direct detection astronomy [60–62].The Partial Rate Matrix for Dark Matter Scattering Benjamin Lillard1,∗ 1Institute for Fundamental Science and Department of Physics, Willamette Hall, University of Oregon, Eugene, OR 97401, U.S.A. (Dated: January 10, 2025) I present a highly efficient integration method for scattering calculations, and a “partial rate matrix”thatencodesthescatteringrateasafunctionoftheSO(3)orientationofthedetector.Iincludeaschemetofullyfactorizethe dark matter particle model, its astrophysical velocity distribution, and the properties of the target material from each other, enabling efficient calculation of the partial rate matrix even in studies comparing large sets of these input functions.The search for low mass (e.g. sub-GeV) R gχ locity distributions; N particle models of the DM, dark matter (DM) is driving a search for detector ma- DM specifying the DM mass m and scattering form factor terials with lower energy thresholds, including semicon- χ F ; and N different models for the detector physics ductors [1–5], molecular targets [6–8], and a variety of DM fS (e.g. for different target materials or final states).Predicting the DM scat- standard approach, the rate integral must be evaluated tering rate in such experiments requires input from con- N N N N many times, which soon becomes pro- densed matter or physical chemistry, complicating what R gχ DM fS hibitively expensive [63].for liquid noble elements [19–24] would have been a rel- In this paper I present a method of “vector space in- atively simple calculation [25].tegration” that factorizes the astrophysics, the DM par- Materialswithdirectionallydependentscatteringrates ticle physics, and the SM physics of the target material havethecapabilitytodistinguishtheDMsignalfromany by projecting the velocity distribution g (⃗v) and the SM χ Standard Model (SM) backgrounds, but at a computa- momentum form factor f2(⃗q) onto bases of orthogonalAfter an initial, numerically intensive stage over the three-dimensional momentum transfer and DM where each object is projected onto its respective vector velocity vectors, ⃗q and ⃗v, rather than simply |⃗q| and |⃗v|.(N = 1), the factorization of N , N and N can R gχ fS DM Therearelargeuncertaintiesontheastrophysicaldark reduce the computational difficulty by multiple orders of matter velocity distribution, which probably contains magnitude.A companion paper [45] introduces a basis computation time that is demonstrated here for direct of spherical wavelets specifically designed for this appli- detection.TheDM–SMscatteringisrepresented ρ is the local DM mass density; µ = m m /(m + χ χ χ SM χ by a matrix M, with coefficients: m ) is the reduced mass of the SM–DM system, forS stM rucp ta ur rt eic fl ue nm cta ioss nm [4S 7M , ; 48a ]n wd hV icce hllS en(⃗q c, oω de) sis tht ehe prd oy pn ea rm tieic s Mφ ≡(cid:28) ϕ(⃗v)(cid:12) (cid:12) (cid:12) F D2 M(q) δ(cid:18) E+ q2 −⃗q·⃗v(cid:19)(cid:12) (cid:12) (cid:12)φ(⃗q,E)(cid:29) , ϕ (cid:12)4πm µ2 2m (cid:12) of the target material, with volume normalization factor χ χ χ (9) V .∆E s, and V cellS is replaced by 2πf s2(⃗q)δ(E−∆E s): I show that R(R) can be evaluated extremely quickly once put into the form: (cid:90) (cid:88) R=N ρ σ¯ d3qd3vg (⃗v)f2(⃗q) T χ 0 χ s R(R)=N ρ σ¯ (cid:88) ⟨ℓm|R|ℓm′⟩K(ℓ) , (11) s T χ 0 mm′ F2 (q) (cid:18) q2 (cid:19) ℓ,m,m′ × DM δ ∆E + −⃗q·⃗v .K(ℓ) ≡ P min g f2 F2 , (12) mm′ 2µ2m ℓ v ℓm ℓm′ DM I project the functions of ⃗v and (⃗q,E) onto vector χ χ spaces spanned by complete and orthogonal basis func- where g (|⃗v|) = ⟨ℓm|g ⟩ and f2 (|⃗q|,E) = ⟨ℓm′|f2⟩ tions |ϕ⟩ and |φ⟩: ℓm χ ℓm′ S are the projections of g (⃗v) and f2(⃗q,E) onto spherical χ S |ϕ (⃗v)⟩=r(v)(v)Y (vˆ), (3) harmonic modes, and v min(q,E)=E/q+q/(2m χ).(10): it must be integrated over Y (Ω)≡ Y0(Ω) for m=0, (5) ⃗q and ⃗v (and possibly E), for every pair of (ϕ,φ) basis ℓm ℓ √ 2ReYm(Ω) for m>0, functions.States of different radial index n,n′ are mixed by I, and where P is the ℓth Legendre polynomial, and v (q,E) the rotation matrices G(ℓ) mix the m,m′ of fixed ℓ; but ℓ min is defined in the usual way, the different ℓ modes do not mix.Asabriefdemonstration,Fig.1showsthree m′m versions of (cid:12) (cid:12)f S2(cid:11) using only the N q =300,103,104 largest the scattering rate for a rotated detector is given by coefficients (respectively), selected from a much larger initial set with ℓ = 36 and N(q) = 210.The vector space integration method works because the scattering rate integrand depends linearly on the functions g and f2.1: Demonstrating the convergence of a basis func- the scattering operator F2 δ(...) is block-diagonal in tion expansion f2 ≃ (cid:80) ⟨φ|f2⟩|φ⟩, summed over the N = DM s φ s a spherical harmonic basis.Acknowledgments Using the partial rate matrix K(ℓ) , the direct de- mm′ tection calculation becomes several orders of magnitude I thank Patrick Draper for collaboration in an early faster.This work was supported in part by the determine how variations in g χ affect the DM scattering U.S. Department of Energy under Grant Number DE- rates.[45] introduces orthogonal spherical wavelets atedratecalculationmakesitmucheasiertoconductthe h (x) defined on x∈[0,1], statistical analyses needed to extract information about λµ theparticlemodel(m ,F )fromthedata.(A7) n=0 To quantify the rates of convergence of the basis func- Three dimensional functions f(⃗u) of finite range |⃗u| ≤ tion expansion, Fig.1 uses a simple particle-in-a-box u can be expanded in a spherical wavelet-harmonic max model to generate a set of analytic detector response basis, functions,borrowedfromthecompanionpaper[45].[15] A. Coskuner, T. Trickle, Z. Zhang, and K. M. Zurek, [47] T. Trickle, Z. Zhang, K. M. Zurek, K. Inzani, and S. M. Phys.This favorable result is not quoted in R gχ DM fS scaling of the direct detection rate calculations.".You have three tasks, which are:\n    1.to summarize the text I provided into a Summary .Please answer within 150-300 characters.\n    2.to summarize the text I provided, using up to seven Highlight.\n    3.to summarize the text I provided, using up to seven Key Insights. Each insight should include a brief in-depth analysis. Key Insight should not include timestamps.\n    Your output should use the following template strictly, provide the results for the three tasks:\n    ## Summary\n    ## Highlights\n    - Highlights\n    ## Key Insights\n    - Key Insights .\n  Importantly your output must use language "English"
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