WIMP-Nucleus Scattering

WIMP-Nucleus Scattering 

Gary Prézeau 

(Caltech) 

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◦ The nature of dark matter remains a 

standing problem in cosmology and 

particle physics. 

◦ It’s relic abundance can be used to 

determine the scale of its interactions. 

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Ω χ ~3x10 -27 cm 3 s -1 /~10 -1 

~10 -26 cm 3 s -1 

About the weak scale 

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◦ Thus, a particle that interacts 

weakly looks like a promising dark 

matter candidate (e.g. neutralinos, 

KK modes, sterile neutrinos). 

◦ There is also a chance of detecting 

it directly by looking at the recoil of 

a nucleus that scattered a WIMP. 

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◦ Can consider the WIMP in a 

particular model, such as MSSM 

◦ In that case, you know all the 

couplings and the calculation of 

amplitudes (like scattering) is 

straightforward 

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◦ In particular, couplings of 

neutralinos to quarks and gluons 

can be expressed in terms of the 

MSSM parameters 

◦ From there, spin-dependent (SD) 

and spin-independent (SI) WIMPparton 

interactions can be derived. 

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At tree level, the following diagrams contribute to 

the effective SI neutralino-quark coupling: 

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SI neutralino-quark 

effective coupling 

after expanding in 

inverse powers of the 

heavy masses: 

C 1 qqχχ 

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From M. Drees and M. Nojiri Phys. Rev. D 

47, 4226–4232 (1993) the following 

diagrams contribute to the SI neutralinogluon 

coupling: 

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SI neutralino-gluon effective 

coupling (after expansion): 

C 2 qqG µν,a G µν 

a 

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• From these neutralino-parton 

interactions, you must now 

construct the effective neutralinohadron 

interactions. Typically, this 

means the neutralino-nucleon 

vertex: 

χ 

χ 

N 

N 

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◦ Traditionally, the NNχχ vertex is the 

only one considered, but there 

could be other hadrons that 

contribute to the SI neutralinonucleus 

cross-section. 

◦ What about: ππχχ 

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◦ You need some way to estimate the 

potential size of any new 

contributions relative to NNχχ 

◦ Effective Field Theory (EFT) can 

help. 

◦ More general than MSSM. 

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◦ Use symmetry properties to relate underlying 

model to low-energy Lagrangian. 

L(q,χ) 

SU(2) L SU(2) R , CP, P 

L(N,π,χ) 

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◦ No specific model required. Can write 

most general Lagrangian. 

◦ Instead of MSSM neutralino, can use a 

general WIMP candidate with arbitrary 

quantum numbers. 

◦ WIMP-quark interactions are set by a 

heavy scale > 100GeV 

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◦ The series of possible quark operators 

is truncated at leading order (LO) in 

inverse powers of the heavy scale Λ. 

◦ The series of corresponding hadron 

operators is truncated at some order 

in p/Λ h where p~m π and Λ h ~1GeV 

(Chiral Perturbation Theory). 

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◦ EFT can be used on a wide range of 

physical problems where the 

fundamental interactions are unknown, 

for example, 0νββ and WIMP-nucleus 

scattering. 

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◦ For 0νββ consider (GP, P. Vogel, M. Ramsey- 

Musolf Phys.Rev.D68:034016,2003) 

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◦ Let us look at the application of EFT to WIMPnucleus 

scattering in more detail (GP, A. 

Kurylov, M. Kamionkowski, and P. Vogel Phys. 

Rev. Lett. 91:231301,2003) : 

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◦ This Lagrangian will give rise to ππχχ, 

πN 2 χχ, N 2 χχ vertices that will contribute 

to the scattering amplitude: 

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New


◦ The new diagrams generated by 

these new interactions are: 

Sub-leading 

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◦ As an explicit example, consider the ππχχ 

Lagrangian to NLO (one derivative): 

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◦ Expressions for the coefficients in terms of 

the parameters appearing in the quarkneutralino 

Lagrangian can be derived using 

PCAC and CVC theorems: 

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◦ Leading to: 

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To leading order in p/Λ h , you can generally neglect 

compared to: 

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◦ So far, we have not assumed anything about 

the underlying model. In MSSM however 

X 

X 

X 

X 

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◦ So far, we have not assumed anything about 

the underlying model. In MSSM however 

X 

] 

SI interaction 

Suppressed by p/Λ h 

SD interaction 

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In MSSM we have 

Leading to the SI operators: 

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◦ Thus, the only diagrams that contribute to 

the SI amplitude to LO are: 

+ 

N 

χ 

χ 

N 

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◦ The relative size of these two diagrams can 

be expressed as a ratio: 

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where = F(A) ~ A 

and r depends on the MSSM parameters and 

depends on the two-nucleon matrix 

element.


◦ The pion exchange diagram has a non-trivial 

dependence on nuclear structure. It could 

change from nucleus to nucleus. 

◦ Opposite signs between the two diagrams 

could have a significant effect on detection 

rates from different target nuclei because of 

cancellations that would occur for one 

nucleus and not another. 

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Summary: 

◦ Effective field theory is a powerful tool when 

estimating the relative size of WIMP-hadron 

contributions to scattering amplitudes for arbitrary 

DM models. 

◦ Generally, one can obtain large contributions from 

pion-exchange to SI amplitude. 

◦ Detection rates could depend non-trivially on 

target nuclei. 

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WIMP-Nucleus Scattering

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