Crystal responses to general dark matter-electron interactions

Crystal responses to general dark matter-electron interactions

Phys.Rev.Res. 3 (2021) 3, 033149 (arXiv: 2105.02233)

Einar Urdshals August 20, 2021

Chalmers

Introduction

• We model dark matter (DM) electron interactions in Silicon and Germanium crystals used in dark matter direct detection experiments.

p p'=p-q

χ χ

e^- E1<0

E2<0

inelas�c

(b)excita�on

• These interactions are interesting because they can be probed for DM particles as light as a fraction of an MeV due to the small mass of the electron.

• So far this scattering has only been considered for simple models such as the dark photon model. We are doing it for arbitrary DM models for the first time using effective non-relativistic operators.

1

Introduction

• We model dark matter (DM) electron interactions in Silicon and Germanium crystals used in dark matter direct detection experiments.

p p'=p-q

χ χ

e^- E1<0

E2<0

inelas�c

(b)excita�on

• These interactions are interesting because they can be probed for DM particles as light as a fraction of an MeV due to the small mass

• So far this scattering has only been considered for simple models such as the dark photon model. We are doing it for arbitrary DM models for the first time using effective non-relativistic operators.

Introduction

• We model dark matter (DM) electron interactions in Silicon and Germanium crystals used in dark matter direct detection experiments.

p p'=p-q

χ χ

e^- E1<0

E2<0

inelas�c

(b)excita�on

• These interactions are interesting because they can be probed for DM particles as light as a fraction of an MeV due to the small mass of the electron.

• So far this scattering has only been considered for simple models such as the dark photon model. We are doing it for arbitrary DM models for the first time using effective non-relativistic operators.

1

Effective operators

O1=1^χe O9=iSχ·

Se×_m^q

e

O3=iSe· _q

m_e×v^⊥_el

O10=iSe·_m^q

e

O4=Sχ·Se O11=iSχ·_m^q

e

O5=iSχ·_q

m_e ×v^⊥_el

O12=Sχ· Se×v_el^⊥ O6=

Sχ· _m^q

e Se·_m^ˆq

e

O13=i Sχ·v^⊥_el Se· _m^q

e

O7=Se·v_el^⊥ O14=i

Sχ·_m^q

e

Se·v^⊥_el O8=S_χ·v^⊥_el O15=iO11

h S_e×v^⊥_el

·_m^q

e

i

Dark matter and crystal responses

Rcrystal= n_χN_cell 128πm²_χm_e²

Z

d(ln ∆E) Z

dq qηb(q,∆E)

×

5

X

l=1

< R_l^∗(q,v)W_l(q,∆E) ,

Catena, Emken, Matas, Spaldin, Urdshals:Phys.Rev.Res. 3 (2021) 3, 033149

• R_l(q,v) is the dark matter response function

• Wl(q,∆E) is the crystal response function

• For the dark photon modelR1(q,v) = 1 orR1(q,v) =α⁴m⁴_e/q⁴, and there is only one contributing crystal response, W₁(q,∆E)

3

Dark matter and crystal responses

Rcrystal= n_χN_cell 128πm²_χm_e²

Z

d(ln ∆E) Z

dq qηb(q,∆E)

×

5

X

l=1

< R_l^∗(q,v)W_l(q,∆E) ,

Catena, Emken, Matas, Spaldin, Urdshals:Phys.Rev.Res. 3 (2021) 3, 033149

• R_l(q,v) is the dark matter response function

• Wl(q,∆E) is the crystal response function

• For the dark photon modelR1(q,v) = 1 orR1(q,v) =α⁴m⁴_e/q⁴, and there is only one contributing crystal response, W₁(q,∆E)

Dark matter and crystal responses

Rcrystal= n_χN_cell 128πm²_χm_e²

Z

d(ln ∆E) Z

dq qηb(q,∆E)

×

5

X

l=1

< R_l^∗(q,v)W_l(q,∆E) ,

Catena, Emken, Matas, Spaldin, Urdshals:Phys.Rev.Res. 3 (2021) 3, 033149

• R_l(q,v) is the dark matter response function

• Wl(q,∆E) is the crystal response function

• For the dark photon modelR1(q,v) = 1 orR1(q,v) =α⁴m⁴_e/q⁴, and there is only one contributing crystal response, W₁(q,∆E)

3

The crystal response W

Wl(q,∆E) = (4π)²Vcell

∆E q²

X

∆Gii⁰

Z

BZ

d³k (2π)³

Z

BZ

d³k⁰ (2π)³Bl

×δ(|k−∆G−k⁰| −q)δ(∆E−Eik+Ei⁰k⁰)

• B₁= f_i,k→i⁰ 0,k⁰

2

• B2=_m^q

e ·(f_i,k→i⁰ 0,k⁰)(f_i,k→i⁰ 0,k⁰)^∗

• B3= f_i,k→i⁰ 0,k⁰

2

• B4=

q

m_e ·f_i,k→i⁰ 0,k⁰

2

• B5=i_m^q

e ·

f_i,k→i⁰ 0,k⁰× f_i,k→i⁰ 0,k⁰

∗

• f_i,k→i⁰ 0,k⁰∼R

d³xψ_i^∗0,k⁰(x)e^ix·qψi,k(x)

• f_i,k→i⁰ 0,k⁰ ∼R

d³xψ^∗_i0,k⁰(x)e^ix·qi∇_m^x

eψi,k(x)

Catena, Emken, Matas, Spaldin, Urdshals:Phys.Rev.Res. 3 (2021) 3, 033149

The crystal response W

Wl(q,∆E) = (4π)²Vcell

∆E q²

X

∆Gii⁰

Z

BZ

d³k (2π)³

Z

BZ

d³k⁰ (2π)³Bl

×δ(|k−∆G−k⁰| −q)δ(∆E−Eik+Ei⁰k⁰)

• B₁= f_i,k→i⁰ 0,k⁰

2

• B2=_m^q

e ·(f_i,k→i⁰ 0,k⁰)(f_i,k→i⁰ 0,k⁰)^∗

• B3= f_i,k→i⁰ 0,k⁰

2

• B₄=

q

m_e ·f_i,k→i⁰ 0,k⁰

2

• B5=i_m^q

e ·

f_i,k→i⁰ 0,k⁰× f_i,k→i⁰ 0,k⁰

∗

• f_i,k→i⁰ 0,k⁰∼R

d³xψ_i^∗0,k⁰(x)e^ix·qψi,k(x)

• f_i,k→i⁰ 0,k⁰ ∼R

d³xψ^∗_i0,k⁰(x)e^ix·qi∇_m^x

eψi,k(x)

Catena, Emken, Matas, Spaldin, Urdshals:Phys.Rev.Res. 3 (2021) 3, 033149

4

The crystal response W

Wl(q,∆E) = (4π)²Vcell

∆E q²

X

∆Gii⁰

Z

BZ

d³k (2π)³

Z

BZ

d³k⁰ (2π)³Bl

×δ(|k−∆G−k⁰| −q)δ(∆E−Eik+Ei⁰k⁰)

• B₁= f_i,k→i⁰ 0,k⁰

2

• B2=_m^q

e ·(f_i,k→i⁰ 0,k⁰)(f_i,k→i⁰ 0,k⁰)^∗

• B3= f_i,k→i⁰ 0,k⁰

2

• B₄=

q

m_e ·f_i,k→i⁰ 0,k⁰

2

• B5=i_m^q

e ·

f_i,k→i⁰ 0,k⁰× f_i,k→i⁰ 0,k⁰

∗

W

Figure 1: W1arising fromB1= f_i,k→i⁰ 0,k⁰

2

Catena, Emken, Matas, Spaldin, Urdshals:Phys.Rev.Res. 3 (2021) 3, 033149

5

<(W

)

Figure 2: <(W2) arising fromB2=_m^q

e·(f_i,k→i⁰ 0,k⁰)(f_i⁰_,k→i0,k⁰)^∗

W

Figure 3: W3arising fromB3= f_i,k→i⁰ 0,k⁰

2

Catena, Emken, Matas, Spaldin, Urdshals:Phys.Rev.Res. 3 (2021) 3, 033149

7

W

Figure 4: W4arising fromB4=

q

m_e·f_i,k→i⁰ 0,k⁰

2

Expected excitation rates

1 2 3 4 5 6 7 8 9 10

Q 10¹⁰

10⁸ 10⁶ 10⁴ 10² 10⁰

[g1day1]

Silicon Anapole c_s8= 8emem g/² cs9= 8emem g/² g/²= 1GeV² m = 10MeV

1(2)

3 crystal

10⁰ 10¹ 10² 10³

m [MeV]

10¹ 10⁰ 10¹ 10² 10³ 10⁴ 10⁵

g/2[GeV2]

Anapole

SENSEI EDELWEISS XENON10 XENON1T DarkSide-50

Figure 5: Rates and limits for anapole interactions,L = ¹_2Λ^g₂χγ¯ ^µγ⁵χ ∂^νFµν

corresponding toc₈^s= 8ememχg

Λ² andc₉^s=−8ememχg Λ²

Catena, Emken, Matas, Spaldin, Urdshals:Phys.Rev.Res. 3 (2021) 3, 033149

9

Summary

• We model general DM electron interaction in Silicon and Germanium crystals due the possibility to probe small dark matter masses.

• In this work, we derive and compute for the first time dark matter and crystal responses that can be used to model any interaction that can be probed in direct detection experiments using Silicon and Germanium crystals.