PHY326/426:Lecture 11
Towards WIMP direct detection
WIMP Cross Sections and Recoil Rates (1)
•Introduction to SUSY dark matter
•WIMP-nucleon collision kinematics
•Recoil energy in the CM frame
•Probability density for recoil energy
Homework (1)
•
•
•
•
Deadline: Mon 17th Nov
•
For PHY426 -25% of overall mark for the module
is on homework 1 and 2 together
Section 1 for PHY326
Section 1 AND 2 for PHY426
For PHY326 -15% of overall mark for the module
is on homework 1 and 2 together
The New Particle Zoo
Here are a few of the candidates on a plot showing cross section
vs. mass. An enormous range. We will focus on WIMPs
thanks to L. Roszkowski (Sheffield)
Why Like Relic CDM Particles?
Indicates Weak Scale
So cosmology indicates generic WIMPs at W&Z scale that
give about the correct value of Ω we need for dark matter:
Candidate 1 (SUSY WIMPs, LSP, Neutralino)
Candidate 2 (UED WIMPs, LKP)
WIMP Candidate 1
Supersymmetric Dark Matter
Each particle gets a “sparticle” counterpart. Bosons get
fermions and vice versa.
e.g. Photon
Photino
W
Wino
Z
Zino etc
The Lightest Supersymmetric Particle (LSP) is predicted to
be stable. This is called the NEUTRALINO.
The Neutralino
The neutralino is a possibility for the lightest
supersymmetric particle, and hence a WIMP candidate.
The neutralino is a quantum mechanical superposition
of the bino, wino and two higgsinos
More in a later lecture on SUSY and dark matter
by Dr. Joel Klinger
How do WIMPs interact?
HALO WIMP
WIMP
rest energy,
10-1000 GeV.
Velocity about 220km/s
ELASTIC SCATTERING
OF A WIMP OFF
A NUCLEUS
NUCLEUS
rest energy,
10-200 GeV.
At rest in lab.
VERY IMPROBABLE FOR
NUCLEUS RECOILS
ANY SINGLE NUCLEUS DUE
TO FEEBLE NATURE OF WEAK INTERACTIONS
Worked Example
Neutralino Dark Matter Properties
Work out the local number density of neutralinos (given that the
local mass density is about 0.3 GeV/c2).
Make the ʻstandardʼ assumption of an isothermal sphere of dark
matter (be aware that this assumption may be an oversimplification, or even wildly incorrect).
ANS: between 0.3 and 30 particles per litre.
Worked Example
Neutralino Dark Matter Properties
Work out the de Broglie Wavelength of a neutralino.
assume velocity ~ 220km s-1 or about 10-3 c.
assume
ANS: the de Broglie wavelength is between 100 and 1 fm.
working given in lecture
The WIMP-induced Recoil Spectrum
So in a detector we want to observe:
(1) a signal - what form does that take?
(2) a background - what form does that take?
The recoil energy spectrum
draw the form of the expected spectrum
given in lecture
How Much Energy can be
Transferred to the Nucleus?
For WIMP mass mW and velocity equal to the virial velocity in our
halo, what is energy of the recoil of the target nucleus?
mW
mT
BEFORE
WIMP
As for any classical two body
elastic collision, maximum
energy transfer when the two
bodies collide head-on.
NUCLEUS
mW
AFTER
Conservation of momentum:
rearrange and square:
Conservation of energy:
mT
Kinematics of Head-on Collisions
from last slide:
multiply cons. of energy equ by 2:
multiply by mw :
Substitute in to momentum conservation equation:
Cancel left hand side with 1st term on right hand side:
Rearrange:
Square:
multiply by mT/mW :
Energy Transferred vs. Masses of
Constituents
1
0
0
1
5
Notice that the energy transferred to the kinetic energy of the
recoiling nucleus is optimal when the nuclear mass equals the
WIMP mass, and for a head on collision the WIMP loses all its
kinetic energy. This is the best possible (and highly unlikely) case.
Worked Example
The Best Transferred Energy
How much kinetic energy could be imparted to a target nucleus ?
In an elastic collision, the maximum energy available for
conversion into recoil of the nucleus is the kinetic energy of the
incident WIMP, IF (when you are lucky), the nuclear mass exactly
equals the WIMP mass.
ANS
working given in lecture
It is very hard to detect this amount of recoil energy in a single
recoiling nucleus embedded in a huge number of other nuclei.
How do you go about detecting
WIMPs?
(1) Ionisation Charge
(2) Scintillation Light
(3) Heat Phonons
More in a later lecture on WIMP detection
!
The Observed Recoil Spectrum
In fact the recoil spectrum actually observed in a detector is quite
a complex product of different factors that we will aim to
understand. It can be written in simplified form as:
dR
2
= R0 S(E R )F (E R )I
dE OBS
This is an
important
formula to know
R0 – the total event rate, determines the over signal rate available
S – the spectral function that comes from the interaction kinematics of,
determined by the relative mass of WIMP and target nucleus
F2 – the form factor correction, for high high A nuclei this supresses
the spectrum
I – the interaction type, whether spin-dependent or spin-independent –
determines the event rate depending on whether the target nucleus
has net nuclear spin or not.
The Observed Recoil Spectrum
dR
2
= R0 S(E R )F (E R )I
dE OBS
We will aim to derive this formula....
!
Note in the following ER is the same as ET
(the energy of the recoiling target nucleus)
Breaking Down R(ER)
The real target of this analysis is the RATE for collisions for a
recoil energy between ER and ER+dER. This is given by
is the number of WIMPs incident per unit cross
sectional area of the target per second (the WIMP flux)
and A is the cross sectional area of the target
consider as two parts:
The probability, GIVEN that a collision occurs, that it
results in a target recoil of kinetic energy ER, and
(b) The probability that an elastic collision occurs at all.
(a)
Extra Note
Bayesʼ theorem
In words for our case here this says ʻprobability of WIMP
colliding with a target nucleus in the detector AND resulting
in a recoil in some energy range is equal to the product of :
The probability, GIVEN that a collision occurs, that it
results in a target recoil of kinetic energy ER, and
(b) The probability that an elastic collision occurs at all.
(a)
WIMP-Nucleon Collision Kinematics
WIMP velocity is typically 10-3c, so we can use non-relativistic
dynamics - do the calculation in the CENTRE of MASS FRAME.
a fancy term for physics as seen by an observer who is at rest
with respect to the centre of mass of the experiment.
What is the velocity vc of the centre of mass frame for a WIMP
colliding with a nucleus, relative to the nucleus ?
wimp
MW
CM
v
target
nucleus mass
MT at rest
x
y
Extra Note
Particle Momentum in CM Frame
Momentum of each particle to an observer in the CM frame
TARGET NUCLEUS. Moves at velocity -vc so momentum is -MTvc
or
WIMP. Moves at velocity v-vc so its momentum is MW(v-vc).
Therefore in the centre of mass frame, the momenta of the WIMP
and the target nucleus are equal in magnitude and opposite in sign
Two Body Elastic Collisions in CM
Use centre of mass scattering
centre of mass
scattering angle
Write
Define the reduced mass of the WIMP & TARGET:
so that
and
Conservation of Energy in CM Frame
so
What we need is the ENERGY TRANSFER - the amount of
energy imparted (in the LAB frame) to the nucleus by the WIMP.
In CM frame, the TARGET momentum before collision = -mTvC,
as the only velocity comes from the motion of the centre of mass.
The WIMP momentum before the collision has the same
magnitude, mTvC , as the momentum of the target.
After collision, the magnitude of the momentum stays the same.
Therefore the final velocity of the WIMP is mTvC/mW.
This resolves into two CM frame components. In the lab frame the
horizontal component of the velocity gets an addition of vC.
Final State Velocities in CM Frame
We are interested in the velocity of the target nucleus after the
collision. Its momentum after the collision in the CM frame has the
same magnitude as it did before the collision.
So its velocity resolves into two components, still in the CM frame,
one parallel to the incident direction of the WIMP and the other
perpendicular. In this frame the initial velocity of the target is vC in
the -x direction. Therefore its final velocity has these components:
horizontal CM frame:
vertical CM frame:
Returning to the lab frame, we must subtract the centre of mass
velocity vc from the x component of the target velocity
Final State Target KE in CM Frame
The Kinetic Energy in the lab frame is half the mass of the target
times the sum of the squares of the lab frame velocity components.
Re-write in terms of the reduced mass
and using the calculated value of the centre of mass frame
velocity vc
we get
This is an
important
formula to know
Questions
Work out the de Broglie Wavelength of a neutralino.
Work out the local number density of neutralinos (given that the
local mass density is about 0.3 GeV/c2).
Describe and draw the expected signal recoil spectrum from
WIMP interactions. Why is this form of spectrum going to make
WIMPs difficult to identify?
Describe the form of the equation for the recoil spectrum (4 parts).
Derive an equation that relates the ratio of initial WIMP kinetic
energy to that of the (initially stationary) recoiling target nucleus
after collision in terms of their masses.
Prove by conservation of momentum and energy that the
maximum observable energy from a 10 GeV WIMP interacting in
matter is 4 keV. To get this energy what nucleus would be
needed?
Impact Parameter and Theta
Derive a relation between the impact parameter and the scattering
angle
End