Theory overview of DM-induced phonon excitations Tongyan Lin UCSD - - PowerPoint PPT Presentation

Tongyan Lin
 UCSD

• Fermilab, June 5, 2019

Theory overview of 
 DM-induced phonon excitations

For extended discussion of theory framework, see: 
 TL, 1904.07915 (lecture notes); 
 Griffin, Knapen, TL, Zurek 1807.10291 (crystals); 
 Knapen, TL, Zurek 1611.06228 (superfluid He); 
 additional refs cited throughout this talk

Outline

Why phonons? Calculating DM-phonon excitations Lessons and future work

2

3

• 1. Two most common elementary excitations in solid

state materials: electrons and phonons. Phonons must be considered for low mass dark matter

Momentum transfer

Why phonons?

Energy deposited

1/(interparticle spacing)

q >> O(1-10) keV → recoil against individual nuclei excite phonons (lattice/fluid vibrations),
 most relevant for sub-MeV dark matter q << O(1-10) keV →

*Numbers are material dependent

q < 2mχvmax ∼ 4 keV × (mχ/MeV)

ω < 1

2mχv2 max ∼ 2 eV × (mχ/MeV)

4

• 1. Two most common elementary excitations in solid

state materials: electrons and phonons. Phonons must be considered for low mass dark matter

Why phonons?

ω >> O(0.1) eV → multiphonon excitations, nuclear recoil

excite single phonons (lattice/fluid vibrations),
 most relevant for sub-MeV dark matter

ω << O(0.1) eV →

*Numbers are material dependent

Momentum transfer Energy deposited

q < 2mχvmax ∼ 4 keV × (mχ/MeV)

ω < 1

2mχv2 max ∼ 2 eV × (mχ/MeV)

5

χ χ

φ n n

DM-nucleon scattering DM-phonon scattering

χ χ

ρ

Phonon quasiparticle

Why phonons?

• 1. Two most common elementary excitations in solid

state materials: electrons and phonons. Phonons must be considered for low mass dark matter

Why phonons?

6

keV MeV GeV TeV meV eV keV Dark matter mass Typical recoil energy

Nuclear recoils T

• t

a l D M e n e r g y

Phonons

• 2. Kinematics of phonon excitation is suited to 


~10 keV-MeV dark matter. Phonon energies ~1-100 meV

Why phonons?

7

• 2. Kinematics of phonon excitation is suited to 


~10 keV-MeV dark matter. Phonon energies ~1-100 meV

Energy deposited

Initial DM velocity

q: momentum transfer

0.0 0.2 0.4 0.6 0.8 1.0

q [keV]

50 100 150 200

ω [meV] mX = 100 keV

Massless mediator

Allowed phase space

ω = q2 2mN

Nuclear recoil

ω = q · vi − q2 2mχ

Why phonons?

8

• 2. Kinematics of phonon excitation is suited to 


~10 keV-MeV dark matter. Phonon energies ~1-100 meV

Initial DM velocity

0.0 0.2 0.4 0.6 0.8 1.0

q [keV]

50 100 150 200

ω [meV] mX = 100 keV

Massless mediator

csq

q: momentum transfer

Acoustic phonons

Energy deposited

Allowed phase space

ω = q · vi − q2 2mχ

0.0 0.2 0.4 0.6 0.8 1.0

q [keV]

50 100 150 200

ω [meV] mX = 100 keV

Massless mediator

Why phonons?

• 2. Kinematics of phonon excitation is suited to 


~10 keV-MeV dark matter. Phonon energies ~1-100 meV

Initial DM velocity

q: momentum transfer

Acoustic phonons Optical phonons

Multiphonons
 also possible in
 this phase space

Energy deposited

Allowed phase space

ω = q · vi − q2 2mχ

Why phonons?

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• 3. DM-phonon couplings are material dependent, 


allowing for target & model complementarity

Spin-independent DM-phonon form factor in crystal

|Fν(q)|2 ∝

• X

atoms j

gj q · eν,j(q) e−Wj (q)

√mj

• 2

Phonon branch ν

phonon eigenmodes,
 band structure enters here

Interplay of DM-ion interaction and phonon modes allows for unique excitation spectrum in each crystal, possible background discrimination

DM effective interaction with ion = nucleus + inner shell electrons

gj ≈ gpZj + gn(A − Z)j + geN e,inner

j

Why phonons?

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• 4. Possible directional signal in anisotropic material

Phonon couplings and energies depend on crystal direction. Daily rate modulation as crystal rotates relative to DM wind.

(0, 0, qz) (0, 0, 0) (0, qy, 0)

q

20 40 60 80 100

ω [meV]

Al2O3 phonons

Example band structure

Acoustic Gapped optical modes

Griffin, Knapen, TL, Zurek 1807.10291

Theory framework for calculating
 DM-phonon excitations

12

Acoustic phonon Optical phonon

Dark matter can excite a given phonon when forces on all the ions constructively interfere in the right way. In particle physics language, we have to match interactions with individual ions to the EFT of phonon excitations.

χ

unit cell

X

j0

Dq,j,j0 · eν,j0(q) = ω2

ν,qeν,j(q)

q

Solve eigenvalue problem:

DM-ion interaction

13

Short range potential In Fourier space

σχp = 4πb2

χp

V (r) = 2πbχp gpmχ X

J

gJδ(r − rJ)

V (q) = 2πbχp gpmχ X

J

gJe−iq·rJ

Assuming spin-independent interactions to ion

χ

gJ

• effective coupling

strength between DM and ion (nucleus + inner shell electrons) J

DM-ion interaction

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Long range potential In Fourier space

Need to characterize expectation
 value of this in material

V (q) = 1 q2 X

J

gJe−iq·rJ

V (r) = X

J

gJ |r − rJ|

Assuming spin-independent interactions to ion

χ

gJ

• effective coupling

strength between DM and ion (nucleus + inner shell electrons) J

Dynamic structure factor

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Called the dynamic structure factor for neutron scattering where gJ = AJ

Need more general class of dark matter structure factors depending on models & form of interactions Scattering off a cold target in ground state: Total DM scattering rate:

Γ(vi) ∝ Z d3q σχp(q)S(q, ω)

S(q, ω) ⌘ 1 N X

λf

• X

J

gJhλf|e−iq·rJ|0i

• 2

δ(Eλf ω)

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S(q, ω) =

(0-phonon) + (1-phonon) + (2-phonon) + …

Expansion in and in 
 anharmonic phonon interactions

q2/(MNω)

Phonon comes into play through positions of ions: Quantized displacement field

rJ(t) = r0

J + uJ(t)

uJ(t) ∼ X

q

1 p 2NMJωq

• ˆ

a†

qe∗ qeiωqt + h.c.

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⇠ hq · uJ q · uJ0i

× δ (ω − ων(q))

Dynamic structure factor

Single phonon excitations

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Form factor for spin-independent interaction to excite a phonon in branch ν with momentum q ∝

• X

atoms j

gj q · eν,j(q) e−Wj (q)

√mj

• 2

in unit cell

S(q, ω) ≈ X

ν

1 ων(q)

δ (ω − ων(q))

Debye-Waller factor

≈ 0

Spin-independent interactions of low mass DM excite single longitudinal phonons Which longitudinal phonons get excited depends on gj

[1-phonon]

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q

Longitudinal acoustic (LA) Longitudinal optical (LO)

χ

Dark matter that couples to nucleon number excites acoustic phonons most easily (constructive interference). There is a large suppression in coupling to optical phonons (destructive interference).

unit cell

Single phonon excitations

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DM mass

MeV 10 keV

Optical vs. acoustic

1 2 3

10−43 10−42 10−41 10−40 10−39 10−38 10−37 10−36 10−35 10−34

σn [cm2]

He (multiphonon)

GaAs

ω > 1 meV ω > 1 meV, analytic ω > 25 meV ω > 25 meV, analytic

DM coupling to nucleon number

all projections assume kg-yr exposure Superfluid He: Knapen, TL, Zurek 2017 See also Cox, Melia, Rajendran 1905.05575

massless mediator limit

Acoustic O p t i c a l

DM coupling to charge

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q

Longitudinal acoustic (LA) Longitudinal optical (LO)

E-field

= Oppositely charged ions in crystal

Dark matter that couples to electric charge (such as freeze-in benchmark) excites optical phonons in polar materials

(see talk by S. Griffin)

χ

Knapen, Pyle, TL, Zurek 1712.06598
 Griffin, Knapen, TL, Zurek 1807.10291

Multiphonon excitations

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Schutz and Zurek 1604.08206 
 Knapen, TL, Zurek 1611.06228
 Acanfora, Esposito, Polosa 1902.02361

Less restrictive final space and larger energy deposition can compensate for penalty in emitting extra phonon

q

𝜓

S(q, ω) =

(0-phonon) + (1-phonon) + (2-phonon) + …

Expansion in and in 
 anharmonic phonon interactions

q2/(MNω)

∼ mχv

Multiphonon excitations

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0.5 1.0 1.5 2.0 2.5 3.0 3.5 1 2 3 4 5 6 7 8

log10(S(q, ω) × eV)

−4 −3 −2 −1 1 2 3 4

Resummed calculation from Campbell et al. 2015

Momentum transfer q [keV] Energy ω [meV]

Dynamic structure factor S(q,ω) in superfluid He

1-phonon 2-phonon +
 3-phonon +…

To-do list & wish list

• Single and multiphonon excitations from neutron

scattering or other sources, for calibrating DM signals and detectors?

• Dynamical structure factors in crystals with

multiphonon excitations (in progress w/ S. Knapen,…)

• Promising quasiparticle excitations beyond phonons,

for a variety of spin-independent, spin-dependent (talk by K. Zhang) or other interactions

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Summary

DM-induced excitations of phonons (or any collective excitations in target) are a natural and promising avenue in direct detection of sub-MeV DM. Basic theory ideas are in place, but many different directions to go!

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Thanks!